Counterexample Generation via Per-Theorem Symbolic Verifiers: When Imitation Hurts and Reinforcement Repairs
Paper • 2610.02444 • Published
Datasets:
Tasks:
Text Generation
Formats:
json
Languages:
English
Size:
10K - 100K
ArXiv:
Tags:
mathematics
counterexample-generation
theorem-proving
symbolic-verification
rlvr
reinforcement-learning
License:
theorem_id stringlengths 4 62 | theorem_name stringlengths 11 96 | domain stringclasses 6
values | original_full_statement stringlengths 32 599 | dropped_hypothesis stringlengths 5 184 | dropped_hypothesis_name stringclasses 7
values | remaining_hypotheses listlengths 0 6 | conclusion stringlengths 5 322 | false_conjecture stringlengths 102 586 | false_conjecture_annotated stringlengths 148 632 ⌀ | counterexample_code stringlengths 269 5.45k | verification_result dict | negation_form_validated bool 1
class | attempt_number int64 1 5 | timestamp stringlengths 32 32 | original_theorem_validated null | attempt_status stringclasses 1
value | verifier dict | canonical_witness stringlengths 1 267 ⌀ | witness_passes_verifier bool 2
classes | verifier_generation_attempts int64 1 6 | migration_failed bool 1
class |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
nilpotent_implies_solvable | Solvability of Nilpotent Groups | GroupTheory | Every nilpotent group is solvable. | G is a nilpotent group | h1 | [] | G is a solvable group | Claim: For all mathematical objects satisfying:
-
the following conclusion holds:
G is a solvable group | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
-
the following conclusion holds:
G is a solvable group | from sympy.combinatorics.named_groups import AlternatingGroup
# We need a group that is NOT solvable.
# The alternating group A5 is the smallest non-solvable group.
G = AlternatingGroup(5)
# Remaining hypotheses: G is a group (inherent to AlternatingGroup)
assert G.order() == 60, "A5 should have order 60"
# Conclusi... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nGroup: A5 (order 60)\nA5 is a group, but it is NOT solvable.",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"negation_form_ok": true,
"safety_ok": true,
"error_message": ""
} | true | 1 | 2026-05-10T17:56:08.274218+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/nilpotent_implies_solvable__h1.py",
"condition_labels": [
"¬Q: G is not a solvable group"
],
"n_conditions": 1,
"n_hyp": 0,
"neg_q_index": 0
} | AlternatingGroup(5) | true | 1 | false |
residual_nilpotence_direct_product | Residual nilpotence of direct products | GroupTheory | If G and H are residually nilpotent groups, then their direct product G × H is a residually nilpotent group. | G is a residually nilpotent group | h1 | [
"H is a residually nilpotent group"
] | The direct product G × H is a residually nilpotent group | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- H is a residually nilpotent group
the following conclusion holds:
The direct product G × H is a residually nilpotent group | null | from sympy.combinatorics import SymmetricGroup, PermutationGroup, Permutation
# Mathematical Context:
# For finite groups, being "residually nilpotent" is exactly equivalent to being "nilpotent",
# because the lower central series of a finite group must eventually stabilize.
# If it stabilizes at the trivial subgroup,... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nG = S3 (order 6), residually nilpotent = False\nH = S2 (order 2), residually nilpotent = True\nG x H has order 12, residually nilpotent = False",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"negatio... | true | 1 | 2026-05-03T19:58:01.516851+00:00 | null | null | {
"module_path": "data/verifiers/GroupTheory/residual_nilpotence_direct_product__h1.py",
"condition_labels": [
"H_1: H is a residually nilpotent group",
"¬Q: G × H is not a residually nilpotent group"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | (SymmetricGroup(3), SymmetricGroup(2)) | true | 1 | false |
local_solvability_direct_product | Local solvability of direct products | GroupTheory | If G and H are locally solvable groups, then their direct product G × H is a locally solvable group. | G is a locally solvable group | h1 | [
"H is a locally solvable group"
] | The direct product G × H is a locally solvable group | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- H is a locally solvable group
the following conclusion holds:
The direct product G × H is a locally solvable group | null | from sympy.combinatorics import Permutation, PermutationGroup
from sympy.combinatorics.named_groups import AlternatingGroup, SymmetricGroup
# We are looking for a counterexample to the false conjecture:
# "If H is a locally solvable group, then G x H is a locally solvable group."
# (The dropped hypothesis was that G i... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nG = A5 (solvable: False)\nH = S3 (solvable: True)\nG x H is solvable: False",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"negation_form_ok": true,
"safety_ok": true,
"error_message": ""
} | true | 1 | 2026-05-03T19:58:03.971351+00:00 | null | null | {
"module_path": "data/verifiers/GroupTheory/local_solvability_direct_product__h1.py",
"condition_labels": [
"H_1: H is a locally solvable group",
"¬Q: G x H is not a locally solvable group"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | (AlternatingGroup(5), S3) | true | 1 | false |
thm_polycyclic_direct_product | Direct Product of Polycyclic Groups | GroupTheory | If G and H are polycyclic groups, then their direct product G x H is a polycyclic group. | G is a polycyclic group | h1 | [
"H is a polycyclic group"
] | The direct product G x H is a polycyclic group | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- H is a polycyclic group
the following conclusion holds:
The direct product G x H is a polycyclic group | null | from sympy.combinatorics import Permutation, PermutationGroup, AlternatingGroup, CyclicGroup
def is_polycyclic_finite(group):
"""
For finite groups, being polycyclic is equivalent to being solvable.
A polycyclic group has a subnormal series with cyclic factors, which for
finite groups is exactly the d... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nG = A5 (polycyclic: False)\nH = C2 (polycyclic: True)\nG x H is polycyclic: False",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"negation_form_ok": true,
"safety_ok": true,
"error_message": ""
} | true | 1 | 2026-05-03T19:57:26.172938+00:00 | null | null | {
"module_path": "data/verifiers/GroupTheory/thm_polycyclic_direct_product__h1.py",
"condition_labels": [
"H_1: H is a polycyclic group",
"¬Q: The direct product G x H is not a polycyclic group"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | (AlternatingGroup(5), CyclicGroup(2)) | true | 1 | false |
local_nilpotence_direct_product | Local nilpotence of direct products | GroupTheory | If G and H are locally nilpotent groups, then their direct product G × H is a locally nilpotent group. | G is a locally nilpotent group | h1 | [
"H is a locally nilpotent group"
] | The direct product G × H is a locally nilpotent group | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- H is a locally nilpotent group
the following conclusion holds:
The direct product G × H is a locally nilpotent group | null | import sympy
from sympy.combinatorics.named_groups import SymmetricGroup, CyclicGroup
from sympy.combinatorics.group_constructs import DirectProduct
def is_nilpotent(group):
"""
For finite groups, local nilpotence is equivalent to nilpotence.
This helper safely checks the is_nilpotent attribute whether it'... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nGroup G (S3) nilpotent: False\nGroup H (C2) nilpotent: True\nDirect Product G x H nilpotent: False",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"negation_form_ok": true,
"safety_ok": true,
"err... | true | 1 | 2026-05-03T19:57:47.812921+00:00 | null | null | {
"module_path": "data/verifiers/GroupTheory/local_nilpotence_direct_product__h1.py",
"condition_labels": [
"H_1: H is a locally nilpotent group",
"¬Q: G x H is not a locally nilpotent group"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | (SymmetricGroup(3), CyclicGroup(2)) | true | 1 | false |
word_problem_direct_product | Solvability of the word problem in direct products | GroupTheory | If G and H are finitely generated groups with solvable word problems, then their direct product G × H has a solvable word problem. | G is a finitely generated group | h1 | [
"G has a solvable word problem",
"H is a finitely generated group",
"H has a solvable word problem"
] | The direct product G × H has a solvable word problem | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- G has a solvable word problem
- H is a finitely generated group
- H has a solvable word problem
the following conclusion holds:
The direct product G × H has a solvable word problem | null | import math
class Group:
def __init__(self, name, is_finitely_generated, has_solvable_word_problem):
self.name = name
self.is_finitely_generated = is_finitely_generated
self.has_solvable_word_problem = has_solvable_word_problem
def direct_product(G, H):
"""
Constructs the direct pr... | {
"success": true,
"exit_code": 0,
"stdout": "",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"negation_form_ok": true,
"safety_ok": true,
"error_message": ""
} | true | 1 | 2026-05-03T19:57:56.000731+00:00 | null | null | {
"module_path": "data/verifiers/GroupTheory/word_problem_direct_product__h1.py",
"condition_labels": [
"H_1: G has a solvable word problem",
"H_2: H is a finitely generated group",
"H_3: H has a solvable word problem",
"¬Q: The direct product G x H does not have a solvable word problem"
],
"n_c... | (type('Group', (), {'name': 'G', 'is_finitely_generated': False, 'has_solvable_word_problem': True})(), type('Group', (), {'name': 'H', 'is_finitely_generated': True, 'has_solvable_word_problem': True})()) | true | 1 | false |
amenability_direct_product | Amenability of direct products | GroupTheory | If G and H are amenable groups, then their direct product G × H is an amenable group. | G is an amenable group | h1 | [
"H is an amenable group"
] | The direct product G × H is an amenable group | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- H is an amenable group
the following conclusion holds:
The direct product G × H is an amenable group | null | from sympy.combinatorics import Permutation, PermutationGroup
from sympy.combinatorics.free_groups import free_group, FreeGroup
def is_amenable(group):
"""
Helper function to determine if a group is amenable.
- Finite groups are amenable.
- Free groups on 2 or more generators are not amenable.
- A ... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nH = PermutationGroup([\n (0 1)]) (Amenable: True)\nG = <free group on the generators (a, b)> (Amenable: False)\nG x H = (<free group on the generators (a, b)>, PermutationGroup([\n (0 1)])) (Amenable: False)",
"stderr": "",
"timed_out"... | true | 1 | 2026-05-03T19:57:53.195573+00:00 | null | null | {
"module_path": "data/verifiers/GroupTheory/amenability_direct_product__h1.py",
"condition_labels": [
"H_1: H is an amenable group",
"¬Q: G x H is not an amenable group"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | (free_group("a, b")[0], S3) | true | 1 | false |
thm_locally_finite_direct_product | Direct Product of Locally Finite Groups | GroupTheory | If G and H are locally finite groups, then their direct product G x H is a locally finite group. | G is a locally finite group | h1 | [
"H is a locally finite group"
] | The direct product G x H is a locally finite group | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- H is a locally finite group
the following conclusion holds:
The direct product G x H is a locally finite group | null | # ---------------------------------------------------------
# Counterexample to:
# "If G and H are locally finite groups, then G x H is locally finite."
# when the hypothesis "G is locally finite" is dropped.
# ---------------------------------------------------------
# Let H be the cyclic group Z_2 under addition mod... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nH = Z_2 (locally finite)\nG = Z (not locally finite)\nG x H = Z x Z_2\nThe finite subset [(1, 0)] generates an infinite subgroup.\nThus, G x H is not locally finite.",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_err... | true | 1 | 2026-05-03T19:57:19.279262+00:00 | null | null | {
"module_path": "data/verifiers/GroupTheory/thm_locally_finite_direct_product__h1.py",
"condition_labels": [
"H_1: H is a locally finite group",
"¬Q: G x H is not a locally finite group"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | (FreeGroup("x"), CyclicGroup(2)) | true | 1 | false |
bi_orderability_direct_product | Bi-orderability of direct products | GroupTheory | If G and H are bi-orderable groups, then their direct product G × H is a bi-orderable group. | G is a bi-orderable group | h1 | [
"H is a bi-orderable group"
] | The direct product G × H is a bi-orderable group | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- H is a bi-orderable group
the following conclusion holds:
The direct product G × H is a bi-orderable group | null | import sympy
from sympy.combinatorics import Permutation, PermutationGroup
from itertools import chain, combinations
def is_bi_orderable(G):
"""
Checks if a finite group G is bi-orderable.
A group is bi-orderable if it has a positive cone P such that:
1. P U P^-1 = G \ {e}
2. P \cap P^-1 = empty
... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nG: Z_2 (not bi-orderable)\nH: Trivial group (bi-orderable)\nG x H: Isomorphic to Z_2, which is NOT bi-orderable.",
"stderr": "/tmp/ce_verify_bwga0xoe.py:6: SyntaxWarning: invalid escape sequence '\\ '\n \"\"\"",
"timed_out": false,
"synta... | true | 1 | 2026-05-03T19:58:41.376361+00:00 | null | null | {
"module_path": "data/verifiers/GroupTheory/bi_orderability_direct_product__h1.py",
"condition_labels": [
"H_1: H is a bi-orderable group",
"¬Q: The direct product G x H is not a bi-orderable group"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | (__import__('sympy.combinatorics', fromlist=['']).PermutationGroup(__import__('sympy.combinatorics', fromlist=['']).Permutation(0, 1)), __import__('sympy.combinatorics', fromlist=['']).PermutationGroup(__import__('sympy.combinatorics', fromlist=['']).Permutation(0))) | true | 1 | false |
thm_prufer_rank_direct_product | Direct Product of Groups of Finite Prufer Rank | GroupTheory | If G and H are groups of finite Prufer rank, then their direct product G x H is a group of finite Prufer rank. | G is a group of finite Prufer rank | h1 | [
"H is a group of finite Prufer rank"
] | The direct product G x H is a group of finite Prufer rank | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- H is a group of finite Prufer rank
the following conclusion holds:
The direct product G x H is a group of finite Prufer rank | null | from sympy.combinatorics import SymmetricGroup, PermutationGroup
from sympy.combinatorics.free_groups import FreeGroup
def has_finite_prufer_rank(group):
"""
Evaluates whether a group has finite Prüfer rank.
The Prüfer rank of a group is the supremum of the minimum number
of generators of its finitely... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nGroup H (remaining hypothesis): PermutationGroup([\n (0 1 2),\n (2)(0 1)])\n -> Has finite Prüfer rank? True\nGroup G (dropped hypothesis): <free group on the generators (x, y)>\n -> Has finite Prüfer rank? False\nDirect Product G x H: (... | true | 1 | 2026-05-03T19:57:29.600098+00:00 | null | null | {
"module_path": "data/verifiers/GroupTheory/thm_prufer_rank_direct_product__h1.py",
"condition_labels": [
"H_1: H is a group of finite Prufer rank",
"¬Q: The direct product G x H is not a group of finite Prufer rank"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | (__import__('sympy.combinatorics.free_groups', fromlist=['FreeGroup']).FreeGroup('x, y'), S3) | true | 2 | false |
conjugacy_problem_direct_product | Solvability of the conjugacy problem in direct products | GroupTheory | If G and H are finitely generated groups with solvable conjugacy problems, then their direct product G × H has a solvable conjugacy problem. | G is a finitely generated group | h1 | [
"G has a solvable conjugacy problem",
"H is a finitely generated group",
"H has a solvable conjugacy problem"
] | The direct product G × H has a solvable conjugacy problem | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- G has a solvable conjugacy problem
- H is a finitely generated group
- H has a solvable conjugacy problem
the following conclusion holds:
The direct product G × H has a solvable conjugacy problem | null | import sympy
class AbstractGroup:
"""
A mock class to represent abstract groups and their properties,
since SymPy does not natively support non-finitely generated groups
or undecidability properties like the conjugacy problem.
"""
def __init__(self, name, finitely_generated, solvable_conjugacy)... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nGroup G (Finitely Generated: False, Solvable Conjugacy: True)\nGroup H (Finitely Generated: True, Solvable Conjugacy: True)\nDirect Product: Group G x H (Finitely Generated: False, Solvable Conjugacy: False)\nAll remaining hypotheses hold, but t... | true | 1 | 2026-05-03T19:58:15.028287+00:00 | null | null | {
"module_path": "data/verifiers/GroupTheory/conjugacy_problem_direct_product__h1.py",
"condition_labels": [
"H_1: G has a solvable conjugacy problem",
"H_2: H is a finitely generated group",
"H_3: H has a solvable conjugacy problem",
"¬Q: The direct product G x H does not have a solvable conjugacy ... | (type('G', (), {'has_solvable_conjugacy_problem': True, '__mul__': lambda self, other: type('GxH', (), {'has_solvable_conjugacy_problem': False})()})(), type('H', (), {'is_finitely_generated': True, 'has_solvable_conjugacy_problem': True})()) | true | 1 | false |
thm_virtually_abelian_direct_product | Direct Product of Virtually Abelian Groups | GroupTheory | If G and H are virtually abelian groups, then their direct product G x H is a virtually abelian group. | G is a virtually abelian group | h1 | [
"H is a virtually abelian group"
] | The direct product G x H is a virtually abelian group | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- H is a virtually abelian group
the following conclusion holds:
The direct product G x H is a virtually abelian group | null | from sympy.combinatorics.free_groups import FreeGroup, free_group
def is_virtually_abelian(group):
"""
Checks if a group (or a direct product of groups represented as a tuple)
is virtually abelian.
Mathematical facts used:
1. A free group F_n is virtually abelian if and only if n <= 1.
... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nH = <free group on the generators (x,)> (virtually abelian: True)\nG = <free group on the generators (a, b)> (virtually abelian: False)\nG x H = (<free group on the generators (a, b)>, <free group on the generators (x,)>)\nIs G x H virtually abe... | true | 1 | 2026-05-03T19:57:39.096032+00:00 | null | null | {
"module_path": "data/verifiers/GroupTheory/thm_virtually_abelian_direct_product__h1.py",
"condition_labels": [
"H_1: H is a virtually abelian group",
"¬Q: The direct product G x H is not a virtually abelian group"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | (__import__('sympy.combinatorics.free_groups', fromlist=['free_group']).free_group('a, b')[0], __import__('sympy.combinatorics.free_groups', fromlist=['free_group']).free_group('x')[0]) | true | 2 | false |
non_freeness_direct_products_free_groups | Non-Freeness of Direct Products of Free Groups | GroupTheory | If F_1 and F_2 are non-trivial free groups, then their direct product F_1 x F_2 is not a free group. | F_1 is a non-trivial free group | h1 | [
"F_2 is a non-trivial free group"
] | The direct product F_1 \times F_2 is not a free group. | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- F_2 is a non-trivial free group
the following conclusion holds:
The direct product F_1 \times F_2 is not a free group. | null | import sympy
class AbstractGroup:
"""
A minimal representation of a group's isomorphism class with respect to being a free group.
"""
def __init__(self, name, is_free, rank):
self.name = name
self.is_free = is_free
self.rank = rank
@property
def is_nontrivial(self):
... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nF_1: F_1 (Trivial Group, Free rank 0)\nF_2: F_2 (Free Group of rank 2)\nDirect Product: (F_1 x F_2) (Free Group of rank 2)\nSince F_1 is the trivial group, F_1 x F_2 is isomorphic to F_2, which IS a free group.\nThus, the conclusion 'F_1 x F_2 i... | true | 1 | 2026-05-03T19:56:53.456833+00:00 | null | null | {
"module_path": "data/verifiers/GroupTheory/non_freeness_direct_products_free_groups__h1.py",
"condition_labels": [
"H_1: F_2 is a non-trivial free group",
"¬Q: F_1 \\times F_2 is a free group"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | (0, 2) | true | 1 | false |
thm_left_orderable_direct_product | Direct Product of Left-Orderable Groups | GroupTheory | If G and H are left-orderable groups, then their direct product G x H is a left-orderable group. | G is a left-orderable group | h1 | [
"H is a left-orderable group"
] | The direct product G x H is a left-orderable group | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- H is a left-orderable group
the following conclusion holds:
The direct product G x H is a left-orderable group | null | from sympy.combinatorics import Permutation, PermutationGroup
# A finite group is left-orderable if and only if it is the trivial group.
# Any non-trivial finite group contains elements of finite order (torsion).
# In a left-orderable group, if x > e, then x^n > e for all n > 0.
# Thus, a torsion element x^n = e would... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nGroup H (order 1): left-orderable = True\nGroup G (order 2): left-orderable = False\nDirect product G x H (order 2): left-orderable = False",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"negation_fo... | true | 1 | 2026-05-03T19:57:36.134525+00:00 | null | null | {
"module_path": "data/verifiers/GroupTheory/thm_left_orderable_direct_product__h1.py",
"condition_labels": [
"H_1: H is a left-orderable group",
"¬Q: G x H is not a left-orderable group"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | (S3, S3.center()) | true | 1 | false |
thm_min_condition_direct_product | Direct Product of Groups with the Minimum Condition | GroupTheory | If G and H are groups that satisfy the minimum condition on subgroups (Artinian groups), then their direct product G x H also satisfies the minimum condition on subgroups. | G satisfies the minimum condition on subgroups | h1 | [
"H satisfies the minimum condition on subgroups"
] | The direct product G x H satisfies the minimum condition on subgroups | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- H satisfies the minimum condition on subgroups
the following conclusion holds:
The direct product G x H satisfies the minimum condition on subgroups | null | from sympy.combinatorics import SymmetricGroup
class GroupZ:
"""The additive group of integers Z."""
def __str__(self):
return "Z"
def satisfies_min_condition(self):
# Z has an infinite strictly descending chain of subgroups:
# Z > 2Z > 4Z > 8Z > ...
# Therefore, it... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample successfully found!\nH: FiniteGroup(order=6) -> satisfies min condition: True\nG: Z -> satisfies min condition: False\nG x H: Z x FiniteGroup(order=6) -> satisfies min condition: False",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"... | true | 1 | 2026-05-03T19:57:18.820618+00:00 | null | null | {
"module_path": "data/verifiers/GroupTheory/thm_min_condition_direct_product__h1.py",
"condition_labels": [
"H_1: H satisfies the minimum condition on subgroups",
"¬Q: G x H does not satisfy the minimum condition on subgroups"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | (Z, S3) | true | 2 | false |
direct_product_pgroups | Direct Product of p-Groups | GroupTheory | Let G and H be finite groups, and let p be a prime. The direct product G × H is a p-group if and only if both G and H are p-groups. | G and H are finite groups | h1 | [
"p is a prime number"
] | The direct product G × H is a p-group if and only if both G and H are p-groups. | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- p is a prime number
the following conclusion holds:
The direct product G × H is a p-group if and only if both G and H are p-groups. | null | from sympy.combinatorics import PermutationGroup, Permutation
# Remaining Hypothesis: p is a prime number
p = 2
assert p == 2, "p is a prime number"
def is_pgroup(group, p):
"""
Strict definition of a p-group: a group whose order is a power of p, strictly greater than 1.
Under this strict definition, the ... | {
"success": true,
"exit_code": 0,
"stdout": "--- Counterexample Details ---\nPrime p = 2\nOrder of G: 2 -> is_pgroup: True\nOrder of H: 1 -> is_pgroup: False\nOrder of GxH: 2 -> is_pgroup: True\nBoth G and H are p-groups: False\nConclusion (GxH is p-group iff both are): False",
"stderr": "",
"timed_out": fal... | true | 1 | 2026-05-03T19:56:59.036099+00:00 | null | null | {
"module_path": "data/verifiers/GroupTheory/direct_product_pgroups__h1.py",
"condition_labels": [
"H_1: p is a prime number",
"¬Q: The direct product G x H is a p-group iff both G and H are p-groups fails"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | (2, PermutationGroup([Permutation(0, 1)]), PermutationGroup([Permutation(0)])) | true | 1 | false |
transitive_prime_degree_primitive | Transitive Permutation Group of Prime Degree is Primitive | GroupTheory | If G is a permutation group acting transitively on a set of prime size p, then G acts primitively on the set. | Let p be a prime number | h1 | [
"Let G be a permutation group acting transitively on a set of size p"
] | G acts primitively on the set | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- Let G be a permutation group acting transitively on a set of size p
the following conclusion holds:
G acts primitively on the set | null | from sympy.combinatorics import Permutation, PermutationGroup
# Define the Dihedral group D4 acting on 4 points (vertices of a square)
# r is a 90-degree rotation, s is a reflection.
r = Permutation(0, 1, 2, 3)
s = Permutation(0, 3)(1, 2)
G = PermutationGroup(r, s)
# The size of the set G acts on is p = 4 (which is c... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nGroup G: Dihedral group of order 8 acting on 4 points\nGenerators: [Permutation(0, 1, 2, 3), Permutation(0, 3)(1, 2)]\nIs G transitive? True\nIs G primitive? False\nThe group is transitive but imprimitive (preserves opposite vertices), disprovin... | true | 1 | 2026-05-03T19:36:16.133220+00:00 | null | null | {
"module_path": "data/verifiers/GroupTheory/transitive_prime_degree_primitive__h1.py",
"condition_labels": [
"H_1: G is a permutation group acting transitively",
"¬Q: G does not act primitively"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | D4 | true | 1 | false |
embedding_wreath_products | Embedding of Wreath Products into Symmetric Groups | GroupTheory | The wreath product of two symmetric groups S_m \wr S_n is isomorphic to a subgroup of S_{mn}. | m and n are positive integers | h1 | [
"S_m and S_n are the symmetric groups of degree m and n, respectively"
] | The wreath product S_m \wr S_n is isomorphic to a subgroup of the symmetric group S_{mn}. | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- S_m and S_n are the symmetric groups of degree m and n, respectively
the following conclusion holds:
The wreath product S_m \wr S_n is isomorphic to a subgroup of the symmetric group S_{mn}. | null | import math
# The dropped hypothesis is that m and n are POSITIVE integers.
# We construct a counterexample by letting m be a non-negative integer that is NOT positive (i.e., m = 0).
m = 0
n = 2
# Remaining hypothesis: S_m and S_n are the symmetric groups of degree m and n.
# For symmetric groups to be defined, their... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nm = 0, n = 2 (Note: m is not a positive integer)\nOrder of S_0: 1\nOrder of S_2: 2\nOrder of the wreath product S_0 \\wr S_2: 2\nOrder of S_{0*2} = S_0: 1\n\nSince the wreath product has order 2 and S_0 has order 1,\nS_0 \\wr S_2 cannot be isomo... | true | 1 | 2026-05-03T19:35:48.036003+00:00 | null | null | {
"module_path": "data/verifiers/GroupTheory/embedding_wreath_products__h1.py",
"condition_labels": [
"H_1: S_m and S_n are the symmetric groups of degree m and n, respectively",
"¬Q: The wreath product S_m \\wr S_n is NOT isomorphic to a subgroup of S_{mn}"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_... | (0, 2) | true | 1 | false |
orbits_normal_subgroup_block_system | Orbits of a Normal Subgroup form a Block System | GroupTheory | If G is a transitive permutation group acting on a set \Omega and N is a normal subgroup of G, then the orbits of N form a system of imprimitivity (block system) for G. | G is a permutation group acting transitively on a set \Omega | h1 | [
"N is a normal subgroup of G"
] | The orbits of N on \Omega form a system of imprimitivity (block system) for the action of G. | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- N is a normal subgroup of G
the following conclusion holds:
The orbits of N on \Omega form a system of imprimitivity (block system) for the action of G. | null | import sympy
from sympy.combinatorics import Permutation, PermutationGroup
# We construct an intransitive permutation group G acting on \Omega = {0, 1, 2, 3, 4}.
# Let G be the direct product of S_3 (acting on {0, 1, 2}) and S_2 (acting on {3, 4}).
p1 = Permutation([1, 2, 0, 3, 4]) # (0 1 2)
p2 = Permutation([1, 0, 2... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found successfully!\nOrbits of N on \\Omega: [{0, 1, 2}, {3, 4}]\nOrbit sizes: {2, 3} (Not uniform, hence not a block system)",
"stderr": "/tmp/ce_verify_st_ur8o2.py:45: SyntaxWarning: invalid escape sequence '\\O'\n print(f\"Orbits of N on \\Omega: ... | true | 1 | 2026-05-03T19:35:28.329137+00:00 | null | null | {
"module_path": "data/verifiers/GroupTheory/orbits_normal_subgroup_block_system__h1.py",
"condition_labels": [
"H_1: N is a normal subgroup of G",
"¬Q: The orbits of N on \\Omega do not form a block system for G"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | (PermutationGroup(Permutation([1, 2, 0, 3, 4]), Permutation([1, 0, 2, 3, 4]), Permutation([0, 1, 2, 4, 3])), PermutationGroup(Permutation([1, 2, 0, 3, 4]), Permutation([0, 1, 2, 4, 3]))) | true | 1 | false |
recursive_characterization_multiple_transitivity | Recursive Characterization of Multiple Transitivity | GroupTheory | A permutation group G acting on a set \Omega is k-transitive if and only if it is transitive and the stabilizer of a point \alpha \in \Omega acts (k-1)-transitively on \Omega \setminus \{\alpha\}. | G is a permutation group acting on a set \Omega | h1 | [
"k is an integer strictly greater than 1",
"The cardinality of \\Omega is at least k"
] | G is k-transitive on \Omega if and only if G is transitive on \Omega and for any \alpha \in \Omega, the point stabilizer G_\alpha acts (k-1)-transitively on \Omega \setminus \{\alpha\}. | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- k is an integer strictly greater than 1
- The cardinality of \Omega is at least k
the following conclusion holds:
G is k-transitive on \Omega if and only if G is transitive on \Omega and for any \alpha \in \Omega, the ... | null | import itertools
# Remaining hypotheses
k = 2
Omega = {0, 1, 2}
# [1] k is an integer strictly greater than 1
assert isinstance(k, int) and k > 1, "Hypothesis [1] failed"
# [2] The cardinality of \Omega is at least k
assert len(Omega) >= k, "Hypothesis [2] failed"
# G is a set of permutations (bijections) on Omega,... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nDomain Omega: {0, 1, 2}\nk = 2\nG consists of the identity and the 3 transpositions (not a group).\nLHS (G is 2-transitive) = False\nRHS (G is transitive and stabilizers are 1-transitive) = True\nTherefore, LHS <-> RHS is FALSE.",
"stderr": ""... | true | 1 | 2026-05-03T19:35:28.042328+00:00 | null | null | {
"module_path": "data/verifiers/GroupTheory/recursive_characterization_multiple_transitivity__h1.py",
"condition_labels": [
"H_1: k is an integer strictly greater than 1",
"H_2: The cardinality of \\Omega is at least k",
"¬Q: G is k-transitive on \\Omega iff G is transitive and G_\\alpha is (k-1)-trans... | ([{0: 0, 1: 1, 2: 2}, {0: 1, 1: 0, 2: 2}, {0: 2, 1: 1, 2: 0}, {0: 0, 1: 2, 2: 1}], {0, 1, 2}, 2) | true | 1 | false |
sn_permutation_matrices_iso | Isomorphism of the symmetric group and permutation matrices | GroupTheory | The symmetric group S_n is isomorphic to the group of n × n permutation matrices under matrix multiplication. | n is a positive integer | h1 | [
"P_n is the set of all n × n permutation matrices"
] | P_n forms a group under matrix multiplication that is isomorphic to the symmetric group S_n. | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- P_n is the set of all n × n permutation matrices
the following conclusion holds:
P_n forms a group under matrix multiplication that is isomorphic to the symmetric group S_n. | null | import sympy
from sympy.combinatorics import SymmetricGroup
# Dropped hypothesis: n is a positive integer
# Counterexample: n = 0 (a non-positive integer)
n = 0
print(f"Counterexample: n = {n}")
# Remaining Hypothesis: P_n is the set of all n x n permutation matrices
# For n = 0, P_0 is the set containing exactly on... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample: n = 0\nP_n for n=0 is the set containing the 0x0 matrix: [Matrix(0, 0, [])]\nP_n forms a valid group under matrix multiplication.\nS_0 is undefined: IndexError - list index out of range\nConclusion correctly fails!",
"stderr": "",
"timed_out": fals... | true | 2 | 2026-05-03T19:35:57.169524+00:00 | null | null | {
"module_path": "data/verifiers/GroupTheory/sn_permutation_matrices_iso__h1.py",
"condition_labels": [
"H_1: P_n is the set of all n x n permutation matrices",
"¬Q: P_n does not form a group under matrix multiplication isomorphic to S_n"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | -1 | true | 2 | false |
supersolvable_implies_solvable | Solvability of Supersolvable Groups | GroupTheory | Every supersolvable group is solvable. | G is a supersolvable group | h1 | [] | G is a solvable group | Claim: For all mathematical objects satisfying:
-
the following conclusion holds:
G is a solvable group | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
-
the following conclusion holds:
G is a solvable group | from sympy.combinatorics.named_groups import SymmetricGroup
# S5 is the symmetric group on 5 elements, which is the smallest symmetric group that is not solvable
G = SymmetricGroup(5)
# Remaining hypotheses: G is a group (always true for PermutationGroup)
assert G.order() == 120, "G must be a valid group"
# Conclusi... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nGroup: SymmetricGroup(5) (order 120)\nIs solvable: False",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"negation_form_ok": true,
"safety_ok": true,
"error_message": ""
} | true | 1 | 2026-05-10T17:56:11.326486+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/supersolvable_implies_solvable__h1.py",
"condition_labels": [
"¬Q: G is not a solvable group"
],
"n_conditions": 1,
"n_hyp": 0,
"neg_q_index": 0
} | SymmetricGroup(5) | true | 1 | false |
dihedral_groups_solvable | Solvability of Dihedral Groups | GroupTheory | Every dihedral group is solvable. | G is a dihedral group D_n for some integer n >= 3 | h1 | [] | G is a solvable group | Claim: For all mathematical objects satisfying:
-
the following conclusion holds:
G is a solvable group | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
-
the following conclusion holds:
G is a solvable group | from sympy.combinatorics.named_groups import AlternatingGroup
# We need a group G that is not solvable.
# The Alternating Group A5 is the smallest non-solvable group.
G = AlternatingGroup(5)
# Remaining hypothesis: G is a group (implicitly true for AlternatingGroup)
# We verify it's a valid group by checking its orde... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nGroup: A5 (order 60)\nA5 is not solvable, which disproves the conjecture when the dihedral hypothesis is dropped.",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"negation_form_ok": true,
"safety_ok... | true | 1 | 2026-05-10T17:56:12.092249+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/dihedral_groups_solvable__h1.py",
"condition_labels": [
"¬Q: G is not a solvable group"
],
"n_conditions": 1,
"n_hyp": 0,
"neg_q_index": 0
} | AlternatingGroup(5) | true | 1 | false |
non_solvability_symmetric_groups | Non-Solvability of Symmetric Groups of Degree Greater Than Four | GroupTheory | For any integer $n \ge 5$, the symmetric group $S_n$ is not a solvable group. | Let $n$ be an integer such that $n \ge 5$. | h1 | [
"Let $S_n$ be the symmetric group of degree $n$."
] | The group $S_n$ is not solvable. | Claim: For all mathematical objects satisfying:
- Let $S_n$ be the symmetric group of degree $n$.
the following conclusion holds:
The group $S_n$ is not solvable. | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- Let $S_n$ be the symmetric group of degree $n$.
the following conclusion holds:
The group $S_n$ is not solvable. | from sympy.combinatorics import SymmetricGroup
import math
# We choose n = 4 as our counterexample.
# The dropped hypothesis was n >= 5.
n = 4
# Remaining hypothesis: S_n is the symmetric group of degree n.
G = SymmetricGroup(n)
assert G.degree == n
assert G.order() == math.factorial(n)
# The conclusion is: S_n is ... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found: n = 4\nGroup: S_4 (order 24)\nIs S_4 solvable? True",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"negation_form_ok": true,
"safety_ok": true,
"error_message": ""
} | true | 1 | 2026-05-10T17:56:13.868637+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/non_solvability_symmetric_groups__h1.py",
"condition_labels": [
"H_1: n is a positive integer",
"¬Q: S_n is solvable"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | 4 | true | 1 | false |
sylow_abelian_groups | Sylow p-subgroups of Finite Abelian Groups | GroupTheory | If G is a finite abelian group and p is a prime dividing the order of G, then G has exactly one Sylow p-subgroup. | G is a finite abelian group | h1 | [
"p is a prime number",
"p divides the order of G"
] | G has exactly one Sylow p-subgroup | Claim: For all mathematical objects satisfying:
- p is a prime number
- p divides the order of G
the following conclusion holds:
G has exactly one Sylow p-subgroup | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- p is a prime number
- p divides the order of G
the following conclusion holds:
G has exactly one Sylow p-subgroup | from sympy.combinatorics import Permutation, PermutationGroup
from sympy import isprime
# We use the symmetric group S3, which is finite but non-abelian
# S3 has order 6.
r = Permutation([1, 2, 0])
s = Permutation([1, 0, 2])
G = PermutationGroup(r, s)
# We choose p = 2
p = 2
# Check remaining hypotheses
# [1] p is a... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nGroup G: S3 (order 6)\nPrime p: 2\nNumber of Sylow 2-subgroups: 3\nSince there is more than 1 Sylow 2-subgroup, the conclusion is FALSE.",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"negation_form_... | true | 1 | 2026-05-10T17:56:22.416841+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/sylow_abelian_groups__h1.py",
"condition_labels": [
"H_1: p is a prime number",
"H_2: p divides the order of G",
"¬Q: G does not have exactly one Sylow p-subgroup"
],
"n_conditions": 3,
"n_hyp": 2,
"neg_q_index": 2
} | (S3, 2) | true | 1 | false |
sylow_2_subgroups_order_2p | Number of Sylow 2-subgroups in Non-Abelian Groups of Order 2p | GroupTheory | If G is a non-abelian finite group of order 2p where p is an odd prime, then the number of Sylow 2-subgroups of G is exactly p. | p is an odd prime | h1 | [
"G is a non-abelian finite group of order 2p"
] | The number of Sylow 2-subgroups of G is exactly p | Claim: For all mathematical objects satisfying:
- G is a non-abelian finite group of order 2p
the following conclusion holds:
The number of Sylow 2-subgroups of G is exactly p | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- G is a non-abelian finite group of order 2p
the following conclusion holds:
The number of Sylow 2-subgroups of G is exactly p | from sympy.combinatorics import DihedralGroup
# Counterexample: D4, the dihedral group of symmetries of a square
G = DihedralGroup(4)
order = G.order()
# Remaining hypothesis: G is a non-abelian finite group of order 2p
assert not G.is_abelian, "G must be non-abelian"
assert order % 2 == 0, "Order of G must be even (... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nGroup: DihedralGroup(4) of order 8\np = 4\nNumber of Sylow 2-subgroups = 1\nConclusion that number of Sylow 2-subgroups equals p is FALSE (1 != 4)",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"nega... | true | 1 | 2026-05-10T17:56:22.519820+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/sylow_2_subgroups_order_2p__h1.py",
"condition_labels": [
"H_1: G is a non-abelian finite group of order 2p",
"¬Q: The number of Sylow 2-subgroups of G is not exactly p"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | DihedralGroup(4) | true | 1 | false |
non_solvability_alternating_groups | Non-Solvability of Alternating Groups of Degree Greater Than Four | GroupTheory | For any integer $n \ge 5$, the alternating group $A_n$ is not a solvable group. | Let $n$ be an integer such that $n \ge 5$. | h1 | [
"Let $A_n$ be the alternating group of degree $n$."
] | The group $A_n$ is not solvable. | Claim: For all mathematical objects satisfying:
- Let $A_n$ be the alternating group of degree $n$.
the following conclusion holds:
The group $A_n$ is not solvable. | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- Let $A_n$ be the alternating group of degree $n$.
the following conclusion holds:
The group $A_n$ is not solvable. | from sympy.combinatorics import AlternatingGroup
# Counterexample: n = 4 (since the hypothesis n >= 5 was dropped)
n = 4
A_n = AlternatingGroup(n)
# Remaining hypothesis:
# [1] Let A_n be the alternating group of degree n.
assert A_n.degree == n
# Conclusion to negate: The group A_n is not solvable.
# In python, tha... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found: n = 4\nA_4 is solvable, which disproves the conjecture.",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"negation_form_ok": true,
"safety_ok": true,
"error_message": ""
} | true | 1 | 2026-05-10T17:56:18.309411+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/non_solvability_alternating_groups__h1.py",
"condition_labels": [
"H_1: witness is an alternating group A_n (or an integer n >= 1)",
"¬Q: A_n is solvable"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | 4 | true | 1 | false |
principal_factors_solvable_groups | Principal Factors of Finite Solvable Groups | GroupTheory | Every principal factor of a finite solvable group is an elementary abelian p-group for some prime p. | G is a finite solvable group | h1 | [] | Every principal factor of G is an elementary abelian p-group for some prime p | Claim: For all mathematical objects satisfying:
-
the following conclusion holds:
Every principal factor of G is an elementary abelian p-group for some prime p | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
-
the following conclusion holds:
Every principal factor of G is an elementary abelian p-group for some prime p | from sympy.combinatorics import AlternatingGroup
# Construct the Alternating Group A_5, which is the smallest finite non-solvable group
G = AlternatingGroup(5)
# Remaining hypotheses: G is a group.
# This is inherently satisfied by the AlternatingGroup object.
# Verify that G is indeed not solvable (the dropped hypo... | {
"success": true,
"exit_code": 0,
"stdout": "",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"negation_form_ok": true,
"safety_ok": true,
"error_message": ""
} | true | 1 | 2026-05-10T17:56:24.804792+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/principal_factors_solvable_groups__h1.py",
"condition_labels": [
"¬Q: G has a principal factor that is not an elementary abelian p-group (i.e. G is not solvable)"
],
"n_conditions": 1,
"n_hyp": 0,
"neg_q_index": 0
} | AlternatingGroup(5) | true | 1 | false |
sylow_2_subgroups_dihedral_odd_index | Sylow 2-subgroups of Dihedral Groups of Odd Index | GroupTheory | For any odd integer n > 1, the Sylow 2-subgroups of the dihedral group D_{2n} have order 2 and are generated by a single reflection. | n is an odd integer greater than 1 | h1 | [
"G is the dihedral group D_{2n} of order 2n",
"P is a Sylow 2-subgroup of G"
] | P has order 2 and is generated by a single reflection element | Claim: For all mathematical objects satisfying:
- G is the dihedral group D_{2n} of order 2n
- P is a Sylow 2-subgroup of G
the following conclusion holds:
P has order 2 and is generated by a single reflection element | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- G is the dihedral group D_{2n} of order 2n
- P is a Sylow 2-subgroup of G
the following conclusion holds:
P has order 2 and is generated by a single reflection element | from sympy.combinatorics import DihedralGroup
from sympy.ntheory import factorint
# We choose n = 4, which is even, violating the dropped hypothesis that n is odd.
n = 4
# Hypothesis [1]: G is the dihedral group D_{2n} of order 2n
G = DihedralGroup(n)
assert G.order() == 2 * n, "G must be of order 2n"
# Hypothesis [... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nFor n = 4, G is the dihedral group of order 8.\nIts Sylow 2-subgroup P has order 8, which is not equal to 2.\nThus, the conclusion fails when n is not odd.",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false... | true | 1 | 2026-05-10T17:56:21.207470+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/sylow_2_subgroups_dihedral_odd_index__h1.py",
"condition_labels": [
"H_1: G is the dihedral group D_{2n} of order 2n",
"H_2: P is a Sylow 2-subgroup of G",
"¬Q: P does not have order 2 or is not generated by a single reflection element"
],
"n_conditions":... | (DihedralGroup(4), DihedralGroup(4)) | true | 2 | false |
thm_sylow_2_s5 | Structure of Sylow 2-subgroups of S5 | GroupTheory | The Sylow 2-subgroups of the symmetric group S5 are isomorphic to the dihedral group of order 8. | G is the symmetric group on 5 letters, S_5 | h1 | [
"P is a Sylow 2-subgroup of G"
] | P is isomorphic to the dihedral group of order 8 | Claim: For all mathematical objects satisfying:
- P is a Sylow 2-subgroup of G
the following conclusion holds:
P is isomorphic to the dihedral group of order 8 | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- P is a Sylow 2-subgroup of G
the following conclusion holds:
P is isomorphic to the dihedral group of order 8 | from sympy.combinatorics import Permutation, PermutationGroup
from sympy.combinatorics.named_groups import SymmetricGroup
# We choose G to be the symmetric group on 3 letters, S3
G = SymmetricGroup(3)
# We define P as a subgroup of G generated by a single transposition.
# This will be our candidate for the Sylow 2-su... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nGroup G: S3 (order 6)\nSubgroup P: generated by [Permutation(2)(0, 1)] (order 2)\nP is a Sylow 2-subgroup of G.\nHowever, P has order 2, so it is NOT isomorphic to the dihedral group of order 8.",
"stderr": "",
"timed_out": false,
"syntax_... | true | 1 | 2026-05-10T17:56:19.306784+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/thm_sylow_2_s5__h1.py",
"condition_labels": [
"H_1: P is a Sylow 2-subgroup of G",
"¬Q: P is not isomorphic to the dihedral group of order 8"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | (CyclicGroup(2), CyclicGroup(2)) | true | 2 | false |
solvability_order_p2q | Solvability of Groups of Order p^2q | GroupTheory | Let $p$ and $q$ be distinct prime numbers. Any finite group of order $p^2q$ is solvable. | Let $p$ and $q$ be distinct prime numbers. | h1 | [
"Let $G$ be a finite group.",
"Assume the order of $G$ is exactly $p^2q$."
] | $G$ is a solvable group. | Claim: For all mathematical objects satisfying:
- Let $G$ be a finite group.
- Assume the order of $G$ is exactly $p^2q$.
the following conclusion holds:
$G$ is a solvable group. | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- Let $G$ be a finite group.
- Assume the order of $G$ is exactly $p^2q$.
the following conclusion holds:
$G$ is a solvable group. | from sympy.combinatorics import AlternatingGroup
# The dropped hypothesis is that p and q are distinct prime numbers.
# Without this, p and q can be any integers.
# We choose G to be the alternating group A5, which is the smallest non-solvable group.
# Its order is 60, which we can write as p^2 * q with p = 2 and q = ... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nGroup: A5\nOrder: 60\np = 2, q = 15 (p^2 * q = 60)\nIs G solvable? False",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"negation_form_ok": true,
"safety_ok": true,
"error_message": ""
} | true | 1 | 2026-05-10T17:56:32.939327+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/solvability_order_p2q__h1.py",
"condition_labels": [
"H_1: G is a finite group",
"H_2: The order of G is exactly p^2q",
"¬Q: G is not a solvable group"
],
"n_conditions": 3,
"n_hyp": 2,
"neg_q_index": 2
} | (AlternatingGroup(5), 2, 15) | true | 1 | false |
center_of_nonabelian_simple_group | Center of a Non-Abelian Simple Group | GroupTheory | The center of a non-abelian simple group is trivial. | G is a simple group | h1 | [
"G is a non-abelian group"
] | The center Z(G) is the trivial subgroup. | Claim: For all mathematical objects satisfying:
- G is a non-abelian group
the following conclusion holds:
The center Z(G) is the trivial subgroup. | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- G is a non-abelian group
the following conclusion holds:
The center Z(G) is the trivial subgroup. | from sympy.combinatorics.named_groups import DihedralGroup
# Create D4, the dihedral group of order 8 (symmetries of a square)
G = DihedralGroup(4)
# Remaining hypothesis [1]: G is a non-abelian group
assert not G.is_abelian, "Remaining hypothesis failed: G is abelian"
# Falsify the conclusion: The center Z(G) is th... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nGroup: DihedralGroup(4) of order 8\nG is non-abelian: True\nCenter Z(G) order: 2 (non-trivial)\nCenter elements: [Permutation(3), Permutation(0, 2)(1, 3)]",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,... | true | 1 | 2026-05-10T17:56:33.274822+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/center_of_nonabelian_simple_group__h1.py",
"condition_labels": [
"H_1: G is a non-abelian group",
"¬Q: The center Z(G) is not the trivial subgroup"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | DihedralGroup(4) | true | 1 | false |
alternating_group_a4_solvable | Solvability of the Alternating Group A_4 | GroupTheory | The alternating group A_4 of degree 4 is a solvable group. | G is the alternating group A_4 of degree 4 | h1 | [] | G is a solvable group | Claim: For all mathematical objects satisfying:
-
the following conclusion holds:
G is a solvable group | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
-
the following conclusion holds:
G is a solvable group | from sympy.combinatorics import AlternatingGroup
# The dropped hypothesis was "G is the alternating group A_4 of degree 4".
# The remaining conjecture implies that any group G is solvable.
# We use A_5, the alternating group of degree 5, which is the smallest non-solvable group.
G = AlternatingGroup(5)
# Remaining hy... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nGroup: A_5 (order 60)\nIs solvable? False",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"negation_form_ok": true,
"safety_ok": true,
"error_message": ""
} | true | 1 | 2026-05-10T17:56:39.279929+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/alternating_group_a4_solvable__h1.py",
"condition_labels": [
"¬Q: G is not a solvable group"
],
"n_conditions": 1,
"n_hyp": 0,
"neg_q_index": 0
} | AlternatingGroup(5) | true | 1 | false |
thm_sylow_3_s4 | Sylow 3-subgroups of S4 | GroupTheory | The symmetric group on 4 letters, S4, has exactly 4 Sylow 3-subgroups. | G is the symmetric group on 4 letters, S_4 | h1 | [] | The number of Sylow 3-subgroups of G is exactly 4 | Claim: For all mathematical objects satisfying:
-
the following conclusion holds:
The number of Sylow 3-subgroups of G is exactly 4 | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
-
the following conclusion holds:
The number of Sylow 3-subgroups of G is exactly 4 | from sympy.combinatorics import Permutation, PermutationGroup
# Define G as the symmetric group S3
# S3 is generated by a 3-cycle and a 2-cycle
G = PermutationGroup(Permutation(0, 1, 2), Permutation(0, 1))
# Remaining hypotheses: None explicitly (other than G being a finite group)
assert G.order() == 6, "G must be a ... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nGroup: S3 (order 6)\nNumber of Sylow 3-subgroups: 1",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"negation_form_ok": true,
"safety_ok": true,
"error_message": ""
} | true | 1 | 2026-05-10T17:56:39.656567+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/thm_sylow_3_s4__h1.py",
"condition_labels": [
"¬Q: The number of Sylow 3-subgroups of G is not exactly 4"
],
"n_conditions": 1,
"n_hyp": 0,
"neg_q_index": 0
} | SymmetricGroup(3) | true | 1 | false |
sylow_2_subgroups_s4 | Sylow 2-subgroups of the Symmetric Group S4 | GroupTheory | The Sylow 2-subgroups of the symmetric group S_4 are isomorphic to the dihedral group D_8 of order 8. | G is the symmetric group S_4 of degree 4 | h1 | [
"P is a Sylow 2-subgroup of G"
] | P is isomorphic to the dihedral group D_8 of order 8 | Claim: For all mathematical objects satisfying:
- P is a Sylow 2-subgroup of G
the following conclusion holds:
P is isomorphic to the dihedral group D_8 of order 8 | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- P is a Sylow 2-subgroup of G
the following conclusion holds:
P is isomorphic to the dihedral group D_8 of order 8 | from sympy.combinatorics import SymmetricGroup
# We construct G = S_3, the symmetric group on 3 elements.
G = SymmetricGroup(3)
# P is a Sylow 2-subgroup of G.
P = G.sylow_subgroup(2)
# VERIFY REMAINING HYPOTHESES:
# [1] P is a Sylow 2-subgroup of G
# The order of S_3 is 6 = 2^1 * 3.
# A Sylow 2-subgroup must be a s... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nGroup G: S_3 (order 6)\nSylow 2-subgroup P: PermutationGroup([\n (2)(0 1)]) (order 2)\nP is a Sylow 2-subgroup of G, but it is not isomorphic to D_8 since their orders differ.",
"stderr": "",
"timed_out": false,
"syntax_error": false,
... | true | 1 | 2026-05-10T17:56:39.030738+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/sylow_2_subgroups_s4__h1.py",
"condition_labels": [
"H_1: P is a Sylow 2-subgroup of G",
"¬Q: P is not isomorphic to the dihedral group D_8 of order 8"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | (SymmetricGroup(3), SymmetricGroup(3).sylow_subgroup(2)) | true | 1 | false |
thm_sylow_7_simple_168 | Sylow 7-subgroups of a Simple Group of Order 168 | GroupTheory | If G is a simple group of order 168, then G contains exactly 8 Sylow 7-subgroups. | G is a simple finite group | h1 | [
"The order of G is 168"
] | The number of Sylow 7-subgroups of G is exactly 8 | Claim: For all mathematical objects satisfying:
- The order of G is 168
the following conclusion holds:
The number of Sylow 7-subgroups of G is exactly 8 | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- The order of G is 168
the following conclusion holds:
The number of Sylow 7-subgroups of G is exactly 8 | from sympy.combinatorics import Permutation, PermutationGroup
# Create a cyclic group of order 168
# We use a single cycle of length 168 as the generator.
# The array represents the mapping: 0->1, 1->2, ..., 166->167, 167->0
gen = Permutation(list(range(1, 168)) + [0])
G = PermutationGroup([gen])
# Remaining hypothes... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nGroup G is cyclic of order 168\nNumber of elements of order 7: 6\nNumber of Sylow 7-subgroups (n_7): 1\nn_7 is NOT 8, which disproves the conjecture.",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"n... | true | 1 | 2026-05-10T17:56:40.509990+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/thm_sylow_7_simple_168__h1.py",
"condition_labels": [
"H_1: The order of G is 168",
"¬Q: The number of Sylow 7-subgroups of G is not exactly 8"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | CyclicGroup(168) | true | 1 | false |
hall_sylow_complements | Hall's Theorem on Sylow Complements in Solvable Groups | GroupTheory | If G is a finite solvable group and p is a prime dividing the order of G, then G contains a Sylow p-complement. | Let G be a finite solvable group. | h1 | [
"Let p be a prime number dividing the order of G."
] | G contains at least one Sylow p-complement, which is a subgroup of index equal to the highest power of p dividing the order of G. | Claim: For all mathematical objects satisfying:
- Let p be a prime number dividing the order of G.
the following conclusion holds:
G contains at least one Sylow p-complement, which is a subgroup of index equal to the highest power of p dividing the order of G. | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- Let p be a prime number dividing the order of G.
the following conclusion holds:
G contains at least one Sylow p-complement, which is a subgroup of index equal to the highest power of p dividing the order of G. | import sympy.combinatorics as comb
from sympy.ntheory import factorint
# G is A_5, the alternating group on 5 elements (the smallest non-solvable group)
G = comb.AlternatingGroup(5)
p = 2
order_G = G.order()
# Remaining hypothesis [1]: p is a prime number dividing the order of G.
assert factorint(p) == {p: 1}, "p mu... | {
"success": true,
"exit_code": 0,
"stdout": "",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"negation_form_ok": true,
"safety_ok": true,
"error_message": ""
} | true | 1 | 2026-05-10T17:56:40.975369+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/hall_sylow_complements__h1.py",
"condition_labels": [
"H_1: p is a prime number",
"H_2: p divides the order of G",
"¬Q: G does not contain a Sylow p-complement"
],
"n_conditions": 3,
"n_hyp": 2,
"neg_q_index": 2
} | (AlternatingGroup(5), 2) | true | 1 | false |
center_of_dihedral_group_odd | Center of the Dihedral Group (Odd n) | GroupTheory | The center of the dihedral group D_{2n} of order 2n is trivial if n is odd and n > 1. | G is the dihedral group D_{2n} of order 2n | h1 | [
"n is an odd integer greater than 1"
] | The center Z(G) is the trivial subgroup. | Claim: For all mathematical objects satisfying:
- n is an odd integer greater than 1
the following conclusion holds:
The center Z(G) is the trivial subgroup. | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- n is an odd integer greater than 1
the following conclusion holds:
The center Z(G) is the trivial subgroup. | from sympy.combinatorics import Permutation, PermutationGroup
# Remaining hypothesis: n is an odd integer greater than 1
n = 3
assert n % 2 != 0, "n must be odd"
assert n > 1, "n must be greater than 1"
# Since the hypothesis "G is the dihedral group D_{2n}" is dropped,
# G can be any group. We choose a group with a ... | {
"success": true,
"exit_code": 0,
"stdout": "",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"negation_form_ok": true,
"safety_ok": true,
"error_message": ""
} | true | 1 | 2026-05-10T17:56:42.590027+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/center_of_dihedral_group_odd__h1.py",
"condition_labels": [
"H_1: n is an odd integer greater than 1",
"¬Q: The center Z(G) is not the trivial subgroup"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | (3, Z3) | true | 1 | false |
aut_grp_klein_four | Automorphism Group of the Klein Four-Group | GroupTheory | The automorphism group of the Klein four-group is isomorphic to the symmetric group S_3. | V is the Klein four-group | h1 | [] | The automorphism group Aut(V) is isomorphic to the symmetric group S_3. | Claim: For all mathematical objects satisfying:
-
the following conclusion holds:
The automorphism group Aut(V) is isomorphic to the symmetric group S_3. | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
-
the following conclusion holds:
The automorphism group Aut(V) is isomorphic to the symmetric group S_3. | import itertools
from sympy.combinatorics import Permutation, PermutationGroup, SymmetricGroup
def get_automorphisms_order(G):
"""
Computes the order of the automorphism group of a small permutation group G
by brute-forcing all bijections and checking the homomorphism property.
"""
elements = list(... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nV = Z_3 (order 3)\nOrder of Aut(V): 2\nOrder of S_3: 6\nSince their orders differ, Aut(V) is NOT isomorphic to S_3.",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"negation_form_ok": true,
"safety_... | true | 1 | 2026-05-10T17:56:42.802624+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/aut_grp_klein_four__h1.py",
"condition_labels": [
"¬Q: Aut(V) is not isomorphic to S_3"
],
"n_conditions": 1,
"n_hyp": 0,
"neg_q_index": 0
} | CyclicGroup(3) | true | 1 | false |
thm_sylow_non_abelian_simple | Sylow p-subgroups in Non-Abelian Simple Groups | GroupTheory | If G is a non-abelian finite simple group and p is a prime dividing the order of G, then the number of Sylow p-subgroups of G is strictly greater than 1. | G is a non-abelian finite simple group | h1 | [
"p is a prime number dividing the order of G"
] | The number of Sylow p-subgroups of G is strictly greater than 1 | Claim: For all mathematical objects satisfying:
- p is a prime number dividing the order of G
the following conclusion holds:
The number of Sylow p-subgroups of G is strictly greater than 1 | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- p is a prime number dividing the order of G
the following conclusion holds:
The number of Sylow p-subgroups of G is strictly greater than 1 | from sympy.combinatorics import Permutation, PermutationGroup
from sympy.ntheory import isprime
import itertools
# Let G be the cyclic group of order 2 (which is abelian, so we dropped the non-abelian finite simple group hypothesis)
# We represent it as the symmetric group S2
a = Permutation([1, 0])
G = PermutationGro... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nGroup G is S2 (order 2)\nPrime p: 2\nNumber of Sylow 2-subgroups: 1\nSince 1 is not strictly greater than 1, the conclusion fails.",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"negation_form_ok": t... | true | 1 | 2026-05-10T17:56:43.964816+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/thm_sylow_non_abelian_simple__h1.py",
"condition_labels": [
"H_1: p is a prime number dividing the order of G",
"¬Q: The number of Sylow p-subgroups of G is exactly 1"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | (CyclicGroup(2), 2) | true | 1 | false |
maximal_subgroup_index_solvable | Index of Maximal Subgroups in Solvable Groups | GroupTheory | The index of any maximal subgroup of a finite solvable group is a power of a prime. | G is a finite solvable group | h1 | [
"M is a maximal subgroup of G"
] | The index [G : M] is a power of a prime number | Claim: For all mathematical objects satisfying:
- M is a maximal subgroup of G
the following conclusion holds:
The index [G : M] is a power of a prime number | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- M is a maximal subgroup of G
the following conclusion holds:
The index [G : M] is a power of a prime number | from sympy.combinatorics import Permutation, PermutationGroup, AlternatingGroup
from sympy.ntheory import factorint
# G is the alternating group A5, which is finite but NOT solvable
G = AlternatingGroup(5)
# M is the dihedral group D10, generated by a 5-cycle and a product of two disjoint 2-cycles
a = Permutation([[0... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nGroup G: A5 (order 60)\nSubgroup M: D10 (order 10)\nIndex [G : M] = 6\nPrime factorization of index: {2: 1, 3: 1}\nIs index a prime power? False",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"negati... | true | 1 | 2026-05-10T17:56:45.498403+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/maximal_subgroup_index_solvable__h1.py",
"condition_labels": [
"H_1: M is a maximal subgroup of G",
"not Q: The index of M in G is not a power of a prime number"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | (AlternatingGroup(5), PermutationGroup(Permutation(0, 1, 2, 3, 4), Permutation(1, 4)(2, 3))) | true | 1 | false |
symmetric_group_solvable_small_degree | Solvability of Small Symmetric Groups | GroupTheory | The symmetric group S_n is solvable if and only if n <= 4. | G is the symmetric group S_n | h1 | [
"n is an integer such that n <= 4"
] | G is a solvable group | Claim: For all mathematical objects satisfying:
- n is an integer such that n <= 4
the following conclusion holds:
G is a solvable group | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- n is an integer such that n <= 4
the following conclusion holds:
G is a solvable group | from sympy.combinatorics import SymmetricGroup
# Conjecture: For any integer n <= 4 and any group G, G is solvable.
# Counterexample: n = 4, G = S_5 (which is not solvable)
n = 4
G = SymmetricGroup(5)
# Remaining hypothesis: n is an integer such that n <= 4
assert isinstance(n, int), "n must be an integer"
assert n ... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nn = 4\nG = SymmetricGroup(5), order 120\nG is solvable: False",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"negation_form_ok": true,
"safety_ok": true,
"error_message": ""
} | true | 1 | 2026-05-10T17:56:49.214156+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/symmetric_group_solvable_small_degree__h1.py",
"condition_labels": [
"H_1: n is an integer such that n <= 4",
"¬Q: G is not a solvable group"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | (S5, 4) | true | 3 | false |
perfect_simple_group | Commutator Subgroup of a Non-Abelian Simple Group | GroupTheory | Let G be a non-abelian simple group. Then G is a perfect group, meaning its commutator subgroup G' is equal to G. | G is a simple group | h1 | [
"G is non-abelian"
] | The commutator subgroup of G is equal to G | Claim: For all mathematical objects satisfying:
- G is non-abelian
the following conclusion holds:
The commutator subgroup of G is equal to G | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- G is non-abelian
the following conclusion holds:
The commutator subgroup of G is equal to G | from sympy.combinatorics import Permutation, PermutationGroup
# Construct the symmetric group S3
# r is the 3-cycle (0 1 2), s is the transposition (0 1)
r = Permutation([1, 2, 0])
s = Permutation([1, 0, 2])
G = PermutationGroup(r, s)
# Remaining hypothesis: G is non-abelian
assert not G.is_abelian, "G must be a non-... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nGroup G is S3, which has order 6.\nG is non-abelian: True\nThe commutator subgroup G' has order 3, which is not equal to 6.\nTherefore, G is not a perfect group.",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error":... | true | 1 | 2026-05-10T17:56:50.584743+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/perfect_simple_group__h1.py",
"condition_labels": [
"H_1: G is non-abelian",
"¬Q: The commutator subgroup of G is not equal to G"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | SymmetricGroup(3) | true | 1 | false |
thm_generation_simple_sylow | Generation of Simple Groups by Sylow p-subgroups | GroupTheory | Let G be a non-abelian finite simple group and p a prime dividing the order of G. The subgroup generated by the union of all Sylow p-subgroups of G is equal to G. | G is a non-abelian finite simple group | h1 | [
"p is a prime number dividing the order of G"
] | The subgroup generated by the union of all Sylow p-subgroups of G is equal to G | Claim: For all mathematical objects satisfying:
- p is a prime number dividing the order of G
the following conclusion holds:
The subgroup generated by the union of all Sylow p-subgroups of G is equal to G | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- p is a prime number dividing the order of G
the following conclusion holds:
The subgroup generated by the union of all Sylow p-subgroups of G is equal to G | import sympy
from sympy.combinatorics import Permutation, PermutationGroup
from sympy import isprime
# Counterexample: Symmetric group S3
# S3 is a finite group of order 6, but not a simple group.
r = Permutation(0, 1, 2)
s = Permutation(0, 1)
G = PermutationGroup(r, s)
p = 3
# Remaining hypothesis [1]: p is a prime... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nGroup G is S3, order = 6\nPrime p = 3\nNumber of Sylow 3-subgroups = 1\nOrder of subgroup generated by their union = 3\nThe generated subgroup is NOT equal to G.",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error":... | true | 1 | 2026-05-10T17:57:01.315925+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/thm_generation_simple_sylow__h1.py",
"condition_labels": [
"H_1: p is a prime number dividing the order of G",
"¬Q: The subgroup generated by the union of all Sylow p-subgroups of G is not equal to G"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | (SymmetricGroup(3), 3) | true | 1 | false |
thm_coprime_index_sylow_containment | Coprime Index and Sylow Containment | GroupTheory | Let G be a finite group, p a prime, and H a subgroup of G. H contains a Sylow p-subgroup of G if and only if the index [G:H] is not divisible by p. | G is a finite group | h1 | [
"p is a prime number",
"H is a subgroup of G"
] | H contains a Sylow p-subgroup of G if and only if the index [G:H] is not divisible by p | Claim: For all mathematical objects satisfying:
- p is a prime number
- H is a subgroup of G
the following conclusion holds:
H contains a Sylow p-subgroup of G if and only if the index [G:H] is not divisible by p | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- p is a prime number
- H is a subgroup of G
the following conclusion holds:
H contains a Sylow p-subgroup of G if and only if the index [G:H] is not divisible by p | import sympy
# Dropped hypothesis: G is a finite group
# Remaining hypotheses:
# [1] p is a prime number
# [2] H is a subgroup of G
p = 2
assert sympy.isprime(p), "p must be a prime number"
# We use the infinite group G = Z (integers under addition)
# and its subgroup H = 2Z (even integers under addition)
# We will ... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nGroup G = Z (infinite)\nSubgroup H = 2Z\nPrime p = 2\nH contains Sylow p-subgroup {0}: True\nIndex [G:H] = 2\n[G:H] is not divisible by p: False\nBiconditional conclusion holds: False",
"stderr": "",
"timed_out": false,
"syntax_error": fal... | true | 1 | 2026-05-10T17:56:46.217972+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/thm_coprime_index_sylow_containment__h1.py",
"condition_labels": [
"H_1: p is a prime number",
"H_2: H is a subgroup of G",
"¬Q: The equivalence fails"
],
"n_conditions": 3,
"n_hyp": 2,
"neg_q_index": 2
} | null | false | 6 | false |
nielsen_schreier_theorem | Nielsen-Schreier Theorem | GroupTheory | Every subgroup of a free group is itself a free group. | F is a free group | h1 | [
"H is a subgroup of F"
] | H is a free group. | Claim: For all mathematical objects satisfying:
- H is a subgroup of F
the following conclusion holds:
H is a free group. | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- H is a subgroup of F
the following conclusion holds:
H is a free group. | from sympy.combinatorics import Permutation, PermutationGroup
# Let F be the cyclic group of order 2, represented as the symmetric group S_2
p = Permutation(0, 1)
F = PermutationGroup([p])
# Let H be F itself, which is trivially a subgroup
H = F
# Verify remaining hypothesis: H is a subgroup of F
assert H.is_subgrou... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nGroup F: PermutationGroup([\n (0 1)]) (order 2)\nSubgroup H: PermutationGroup([\n (0 1)]) (order 2)\nH is a subgroup of F, but H is NOT a free group because it is a non-trivial finite group.",
"stderr": "",
"timed_out": false,
"synta... | true | 1 | 2026-05-10T17:57:06.598598+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/nielsen_schreier_theorem__h1.py",
"condition_labels": [
"H_1: H is a subgroup of F",
"¬Q: H is not a free group"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | (CyclicGroup(2), CyclicGroup(2)) | true | 1 | false |
cyclic_abelianization_nilpotent | Cyclic Abelianization Implies Cyclic Group for Nilpotent Groups | GroupTheory | If G is a nilpotent group such that its abelianization G/G' is a cyclic group, then G is a cyclic group. | G is a nilpotent group | h1 | [
"The quotient group G/G' is a cyclic group"
] | G is a cyclic group | Claim: For all mathematical objects satisfying:
- The quotient group G/G' is a cyclic group
the following conclusion holds:
G is a cyclic group | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- The quotient group G/G' is a cyclic group
the following conclusion holds:
G is a cyclic group | from sympy.combinatorics.named_groups import SymmetricGroup
from sympy.ntheory import isprime
# We use the Symmetric Group S_3 as our counterexample.
G = SymmetricGroup(3)
# Compute the derived subgroup G' (commutator subgroup)
G_prime = G.derived_subgroup()
order_G = G.order()
order_G_prime = G_prime.order()
# The... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found: S_3\nOrder of G: 6\nOrder of G': 3\nOrder of G/G': 2 (prime, so G/G' is cyclic)\nMaximum element order in G: 3 (less than 6, so G is not cyclic)",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"negatio... | true | 1 | 2026-05-10T17:57:07.864868+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/cyclic_abelianization_nilpotent__h1.py",
"condition_labels": [
"H_1: The quotient group G/G' is a cyclic group",
"¬Q: G is not a cyclic group"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | SymmetricGroup(3) | true | 1 | false |
thm_direct_product_nilpotent | Nilpotency of Direct Products | GroupTheory | The direct product of two nilpotent groups is nilpotent. | G is a nilpotent group | h1 | [
"H is a nilpotent group"
] | The direct product G \times H is a nilpotent group | Claim: For all mathematical objects satisfying:
- H is a nilpotent group
the following conclusion holds:
The direct product G \times H is a nilpotent group | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- H is a nilpotent group
the following conclusion holds:
The direct product G \times H is a nilpotent group | from sympy.combinatorics import SymmetricGroup, CyclicGroup
from sympy.combinatorics.group_constructs import DirectProduct
# We are looking for a counterexample where G is not nilpotent,
# but H is nilpotent, making the direct product G x H not nilpotent.
# G is the symmetric group on 3 elements, which is not nilpote... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nG = SymmetricGroup(3) (is_nilpotent = False)\nH = CyclicGroup(2) (is_nilpotent = True)\nG x H is nilpotent: False",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"negation_form_ok": true,
"safety_ok... | true | 1 | 2026-05-10T17:57:08.391730+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/thm_direct_product_nilpotent__h1.py",
"condition_labels": [
"H_1: H is a nilpotent group",
"¬Q: The direct product G x H is not a nilpotent group"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | (SymmetricGroup(3), CyclicGroup(2)) | true | 1 | false |
central_element_fixed_point_dichotomy | Central Element Fixed Point Dichotomy | GroupTheory | Let G be a group acting transitively on a set X, and let g be an element in the center of G. Then g either fixes every point in X or fixes no points in X. | G is a group acting transitively on a set X | h1 | [
"g is an element in the center of G, Z(G)"
] | The element g either fixes every point in X or fixes no points in X. | Claim: For all mathematical objects satisfying:
- g is an element in the center of G, Z(G)
the following conclusion holds:
The element g either fixes every point in X or fixes no points in X. | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- g is an element in the center of G, Z(G)
the following conclusion holds:
The element g either fixes every point in X or fixes no points in X. | from sympy.combinatorics import Permutation, PermutationGroup
# We define a set X and a group G acting on X.
# Let X = {0, 1, 2}
X = {0, 1, 2}
# Let g be the permutation that swaps 0 and 1, but fixes 2.
# In array form: 0 -> 1, 1 -> 0, 2 -> 2
g = Permutation([1, 0, 2])
# Let G be the group generated by g.
# G acts ... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nSet X: {0, 1, 2}\nGroup G order: 2\nElement g: (2)(0 1)\nFixed points of g: [2]",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"negation_form_ok": true,
"safety_ok": true,
"error_message": ""
} | true | 1 | 2026-05-10T17:57:08.635810+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/central_element_fixed_point_dichotomy__h1.py",
"condition_labels": [
"H_1: g is an element in the center of G, Z(G)",
"¬Q: g fixes some points in X but not all points in X"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | (PermutationGroup([Permutation(0, 1)]), {0, 1, 2}, Permutation(0, 1)) | true | 1 | false |
solvability_external_direct_product | Solvability of the external direct product of finite solvable groups | GroupTheory | If $H$ and $K$ are finite solvable groups, then their external direct product $H \times K$ is a solvable group. | $H$ is a finite solvable group | h1 | [
"$K$ is a finite solvable group",
"$G$ is the external direct product of $H$ and $K$"
] | $G$ is a solvable group | Claim: For all mathematical objects satisfying:
- $K$ is a finite solvable group
- $G$ is the external direct product of $H$ and $K$
the following conclusion holds:
$G$ is a solvable group | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- $K$ is a finite solvable group
- $G$ is the external direct product of $H$ and $K$
the following conclusion holds:
$G$ is a solvable group | from sympy.combinatorics.named_groups import AlternatingGroup, CyclicGroup
from sympy.combinatorics.group_constructs import DirectProduct
# H is a group that is NOT solvable (e.g., A5)
H = AlternatingGroup(5)
# K is a finite solvable group (e.g., C2)
K = CyclicGroup(2)
# G is the external direct product of H and K
G... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nH is A5, solvable: False\nK is C2, solvable: True\nG is H x K, solvable: False",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"negation_form_ok": true,
"safety_ok": true,
"error_message": ""
} | true | 1 | 2026-05-10T17:57:04.795655+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/solvability_external_direct_product__h1.py",
"condition_labels": [
"H_1: K is a finite solvable group",
"H_2: G is the external direct product of H and K",
"¬Q: G is not a solvable group"
],
"n_conditions": 3,
"n_hyp": 2,
"neg_q_index": 2
} | null | false | 6 | false |
automorphism_group_cyclic_order | Automorphism Group of a Finite Cyclic Group Theorem | GroupTheory | Let G be a finite cyclic group of order n. Then the order of the automorphism group of G is equal to phi(n), where phi is Euler's totient function. | G is a cyclic group | h1 | [
"The order of G is a finite integer n"
] | The order of the automorphism group of G is equal to Euler's totient function phi(n) | Claim: For all mathematical objects satisfying:
- The order of G is a finite integer n
the following conclusion holds:
The order of the automorphism group of G is equal to Euler's totient function phi(n) | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- The order of G is a finite integer n
the following conclusion holds:
The order of the automorphism group of G is equal to Euler's totient function phi(n) | import itertools
from sympy.ntheory import totient
# We construct the Klein four-group (V4), which is a non-cyclic finite group of order 4.
# Elements can be represented as integers 0, 1, 2, 3 with bitwise XOR as the group operation.
elements = [0, 1, 2, 3]
n = len(elements)
# Remaining hypothesis: The order of G is ... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nGroup: Klein four-group (V4)\nOrder of G (n): 4\nOrder of Aut(G): 6\nEuler's totient phi(4): 2",
"stderr": "/tmp/ce_verify_ztqkbncq.py:37: SymPyDeprecationWarning: \n\nThe `sympy.ntheory.factor_.totient` has been moved to `sympy.functions.comb... | true | 1 | 2026-05-10T17:57:09.046407+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/automorphism_group_cyclic_order__h1.py",
"condition_labels": [
"H_1: The order of G is a finite integer n",
"¬Q: The order of the automorphism group of G is not equal to Euler's totient function phi(n)"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | V4 | true | 1 | false |
p_group_center_nontrivial | Non-triviality of the Center of a p-group | GroupTheory | Let p be a prime number and let G be a finite group of order p^n for some positive integer n. Then the center of G, Z(G), is non-trivial (i.e., |Z(G)| > 1). | p is a prime number | h1 | [
"n is a positive integer",
"G is a finite group of order p^n"
] | The center of G, Z(G), has order strictly greater than 1 | Claim: For all mathematical objects satisfying:
- n is a positive integer
- G is a finite group of order p^n
the following conclusion holds:
The center of G, Z(G), has order strictly greater than 1 | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- n is a positive integer
- G is a finite group of order p^n
the following conclusion holds:
The center of G, Z(G), has order strictly greater than 1 | from sympy.combinatorics import Permutation, PermutationGroup
# We construct the symmetric group S3, which has order 6.
# S3 is generated by a 3-cycle and a 2-cycle.
r = Permutation([1, 2, 0]) # (0 1 2)
s = Permutation([1, 0, 2]) # (0 1)
G = PermutationGroup(r, s)
# With the hypothesis "p is a prime number" dropp... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nGroup: S3 (order 6)\np = 6, n = 1\nCenter order: 1",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"negation_form_ok": true,
"safety_ok": true,
"error_message": ""
} | true | 1 | 2026-05-10T17:57:09.107364+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/p_group_center_nontrivial__h1.py",
"condition_labels": [
"H_1: n is a positive integer",
"H_2: G is a finite group of order p^n",
"¬Q: The center of G, Z(G), has order <= 1"
],
"n_conditions": 3,
"n_hyp": 2,
"neg_q_index": 2
} | (SymmetricGroup(3), 6, 1) | true | 1 | false |
thm_sylow_normal_p_power_index | Factorization by Sylow p-subgroups and Normal Subgroups of p-power Index | GroupTheory | Let G be a finite group, P a Sylow p-subgroup of G, and N a normal subgroup of G. If the index [G:N] is a power of p, then G = PN. | G is a finite group | h1 | [
"p is a prime number",
"P is a Sylow p-subgroup of G",
"N is a normal subgroup of G",
"The index [G:N] is a power of p"
] | G = PN | Claim: For all mathematical objects satisfying:
- p is a prime number
- P is a Sylow p-subgroup of G
- N is a normal subgroup of G
- The index [G:N] is a power of p
the following conclusion holds:
G = PN | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- p is a prime number
- P is a Sylow p-subgroup of G
- N is a normal subgroup of G
- The index [G:N] is a power of p
the following conclusion holds:
G = PN | import math
import sympy
class SubgroupOfZ:
"""
Represents a subgroup of the additive group of integers, Z.
Every subgroup of Z is of the form nZ for some non-negative integer n.
"""
def __init__(self, n):
self.n = abs(n)
def is_normal(self):
# Z is abelian, so all its ... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nG = 1Z (The group of integers Z)\np = 2\nP = 0Z (Sylow p-subgroup)\nN = 2Z (Normal subgroup of index 2)\nPN = 2Z\nG == PN is False",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"negation_form_ok": t... | true | 1 | 2026-05-10T17:57:09.815503+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/thm_sylow_normal_p_power_index__h1.py",
"condition_labels": [
"H_1: p is a prime number",
"H_2: P is a Sylow p-subgroup of G",
"H_3: N is a normal subgroup of G",
"H_4: The index of N in G is a power of p",
"¬Q: G != PN"
],
"n_conditions": 5,
"n... | null | false | 5 | false |
finitely_generated_module_over_noetherian | Finitely Generated Modules over Noetherian Rings | RingTheory | If R is a commutative Noetherian ring and M is a finitely generated R-module, then M is a Noetherian module. | R is a commutative Noetherian ring | h1 | [
"M is an R-module",
"M is finitely generated over R"
] | M is a Noetherian module (satisfies the ascending chain condition on submodules) | Claim: For all mathematical objects satisfying:
- M is an R-module
- M is finitely generated over R
the following conclusion holds:
M is a Noetherian module (satisfies the ascending chain condition on submodules) | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- M is an R-module
- M is finitely generated over R
the following conclusion holds:
M is a Noetherian module (satisfies the ascending chain condition on submodules) | import sympy as sp
class InfinitePolynomialRing:
"""
Represents the polynomial ring Q[x_0, x_1, x_2, ...] with infinitely many variables.
This ring is commutative but NOT Noetherian.
"""
def get_var(self, i):
return sp.Symbol(f"x_{i}")
class FreeModule:
"""
Represents a free module... | {
"success": true,
"exit_code": 0,
"stdout": "Counterexample found!\nRing R: Q[x_0, x_1, x_2, ...] (Polynomial ring with infinitely many variables)\nModule M: R (Finitely generated by 1 over itself)\nM is not a Noetherian module because it contains an infinite strictly ascending chain of submodules (ideals).",
... | true | 1 | 2026-05-10T17:57:10.062021+00:00 | null | success | {
"module_path": "data/verifiers/RingTheory/finitely_generated_module_over_noetherian__h1.py",
"condition_labels": [
"H_1: M is an R-module",
"H_2: M is finitely generated over R",
"¬Q: M is not a Noetherian module"
],
"n_conditions": 3,
"n_hyp": 2,
"neg_q_index": 2
} | null | false | 6 | false |
frattini_quotient_solvability | Frattini Quotient Solvability Theorem | GroupTheory | A finite group $G$ is solvable if the quotient group $G/\Phi(G)$ is solvable, where $\Phi(G)$ denotes the Frattini subgroup of $G$. | Let $G$ be a finite group. | h1 | [
"Let $\\Phi(G)$ denote the Frattini subgroup of $G$.",
"Assume that the quotient group $G/\\Phi(G)$ is a solvable group."
] | The group $G$ is solvable. | Claim: For all mathematical objects satisfying:
- Let $\Phi(G)$ denote the Frattini subgroup of $G$.
- Assume that the quotient group $G/\Phi(G)$ is a solvable group.
the following conclusion holds:
The group $G$ is solvable. | [FALSE CONJECTURE — dropped hypothesis: 'h1']
Claim: For all mathematical objects satisfying:
- Let $\Phi(G)$ denote the Frattini subgroup of $G$.
- Assume that the quotient group $G/\Phi(G)$ is a solvable group.
the following conclusion holds:
The group $G$ is solvable. | import sympy
class NeumannInfiniteGroup:
"""
Counterexample to the conjecture when the 'finite group' hypothesis is dropped.
By a theorem of B.H. Neumann (1954), any group can be embedded into a group
that has no maximal subgroups.
Let G be such an infinite group into which the alternat... | {
"success": true,
"exit_code": 0,
"stdout": "",
"stderr": "",
"timed_out": false,
"syntax_error": false,
"runtime_error": false,
"negation_form_ok": true,
"safety_ok": true,
"error_message": ""
} | true | 1 | 2026-05-10T17:57:10.662330+00:00 | null | success | {
"module_path": "data/verifiers/GroupTheory/frattini_quotient_solvability__h1.py",
"condition_labels": [
"H_1: G/Phi(G) is solvable",
"¬Q: G is not solvable"
],
"n_conditions": 2,
"n_hyp": 1,
"neg_q_index": 1
} | null | false | 2 | false |
End of preview. Expand in Data Studio
SymCE
4,707 false undergraduate algebra and real-analysis conjectures, each with an executable Python verifier.
Every record is a true theorem with one hypothesis removed. A verifier module decides, for a candidate witness, whether each remaining hypothesis holds and whether the conclusion fails. Verification takes about 130 ms, so the same substrate serves as the reward function during reinforcement learning: SymCE is a training environment, not only a benchmark.
Companion to Counterexample Generation via Per-Theorem Symbolic Verifiers: When Imitation Hurts and Reinforcement Repairs (Findings of EMNLP 2026).
- Paper: https://aclanthology.org/2026.findings-emnlp.2112/
- arXiv: https://arxiv.org/abs/2610.02444
- Project page: https://ce-rlvr.github.io/SymCE/
- Code: https://github.com/ce-rlvr/SymCE
Configurations
| Config | Rows | Contents |
|---|---|---|
corpus |
4,707 | Every retained record: statement, hypotheses, conclusion, dropped hypothesis, canonical witness, verifier pointer |
test |
201 | Held-out counterexample benchmark, stratified by domain |
sft |
1,000 | Supervised fine-tuning split, chat-formatted |
rlvr |
3,506 | RLVR prompt set with the oracle metadata GRPO needs |
theorem_truth |
347 | Calibration probe. These theorems are true; the correct answer is NONE |
witness_schemas |
4,676 | One-line Python-shape description of an acceptable witness per module |
verifiers |
12,376 | Every generated verifier module, with its source |
from datasets import load_dataset
test = load_dataset("ce-rlvr/SymCE", "test")["test"]
print(test[0]["false_conjecture"])
The verifier tree is also shipped as verifiers.tar.gz for local execution:
hf download ce-rlvr/SymCE verifiers.tar.gz --repo-type dataset --local-dir .
tar xzf verifiers.tar.gz # -> verifiers/<Domain>/<theorem_id>__<hyp>.py
Paths inside the records are written as data/verifiers/<Domain>/<file>.py, which
matches the layout of the code repository.
Running a verifier
from src.oracle.sandbox import run_oracle # from the code repository
run_oracle("data/verifiers/GroupTheory/abelian_direct_product__h1.py",
"(SymmetricGroup(3), CyclicGroup(2))", n_conditions=2).conditions
# [True, True] -> the remaining hypothesis holds and the conclusion fails
The module returns one boolean per remaining hypothesis followed by one for the negated conclusion. A witness counts as a counterexample only when all of them are true. No language model judges the answer.
Domains
| Domain | Strict pool | SFT | RLVR | Test | Verifier modules |
|---|---|---|---|---|---|
| GroupTheory | 874 | 366 | 434 | 74 | 4,146 |
| RingTheory | 565 | 237 | 280 | 48 | 3,775 |
| NumberTheory | 392 | 164 | 195 | 33 | 952 |
| LinearAlgebra | 233 | 98 | 115 | 20 | 1,049 |
| FieldTheory | 166 | 70 | 82 | 14 | 1,471 |
| RealAnalysis | 159 | 67 | 79 | 13 | 983 |
Splits
Stratified per domain with a fixed seed (42). The strict pool is the 2,389 records whose teacher-generated canonical witness passes its own verifier; the remaining 2,318 records have a working verifier but no oracle-confirmed witness, and are used only as RLVR prompts.
Verifier reliability
Verifiers are written by a teacher model, so their reliability was measured rather than assumed. Judging verifier source proved unworkable: the code leans on heuristics that are incomplete in theory but correct on the cases that arise. The accept/reject decision on real model outputs was audited instead:
- 177 decisions checked by hand, 97.7 % correct
- 99/101 accepted witnesses genuinely valid (2.0 % false-accept rate)
- 74/76 rejections genuinely correct
- no case traced to a faulty verifier module
The annotations, including all four disputed decisions, are in human_audit/.
Known limitations
Mutations are single-hypothesis drops only, so the corpus does not span the range of mathematical fallacies. Verifiers are bounded by what SymPy and Z3 decide, which excludes topology and higher-order logic. All verifiers come from a single teacher model (Gemini 3.1 Pro); the witness schemas come from a second (DeepSeek). The audit had one annotator, so there is no inter-annotator agreement figure.
Licence
CC BY 4.0. The corpus consists of formal mathematical statements and their hypothesis-dropped mutations. It contains no human-derived text and no personal information.
Citation
@inproceedings{zouak-etal-2026-counterexample,
title = "Counterexample Generation via Per-Theorem Symbolic Verifiers: When Imitation Hurts and Reinforcement Repairs",
author = "Zouak, Omar Farouk and
Boukhalfa, Houssam Eddine and
Lakehal, Soumaya and
Katiyar, Shiv and
Nefti-Meziani, Samia",
editor = "Lee, Hung-Yi and
Dev, Sunipa and
Li, Jessy and
Elliott, Desmond",
booktitle = "Findings of the {A}ssociation for {C}omputational {L}inguistics: {EMNLP} 2026",
month = oct,
year = "2026",
address = "Budapest, Hungary",
publisher = "Association for Computational Linguistics",
url = "https://aclanthology.org/2026.findings-emnlp.2112/",
pages = "45032--45047",
ISBN = "979-8-89176-462-0",
}
- Downloads last month
- 56