Finite groups that are the product of every pair of non-conjugate maximal subgroups are soluble
Abstract
We prove that every finite group that is the product of every pair of its non-conjugate maximal subgroups is soluble, answering Problem 10.34 of the Kourovka Notebook. The almost-simple case was proved by Tikhonenko and Tyutyanov. We treat the remaining minimal-counterexample branch, where the unique minimal normal subgroup has the form S^k, with S nonabelian simple and k >= 2. Two automorphism-stable coordinate subgroup classes satisfying a maximal-supplement criterion produce non-conjugate maximal supplements. If the factorization hypothesis held, their product would impose a p-adic divisibility requirement growing linearly with k, while the quotient contributes only coordinate outer automorphisms and a factor dividing k!. A fixed valuation gap therefore excludes every k >= 2 at once. Suitable subgroup classes are constructed uniformly across the infinite families of finite simple groups using parabolic, torus, and primitive-prime-divisor arguments; stable flag parabolics handle graph fusion, while GAP certificates cover designated finite and sporadic cases. The resulting all-k obstruction provides a reusable mechanism for eliminating direct-power socles in finite-group factorization problems.
Related
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- Maximal ranks of free abelian subgroups of generalized cactus groups associated with finite Coxeter systems
- Orbits of maximal and submaximal dimension for Sylow p-subgroups of finite classical groups
- Finite Groups with Few Almost Maximal Subgroups
- On c-Normal Maximal Chains of Sylow Subgroups in Finite Groups
BibTeX
@misc{sater2026finite,
title = {{Finite groups that are the product of every pair of non-conjugate maximal subgroups are soluble}},
author = {Richie Sater},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.02970},
eprint = {2608.02970},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.02970},
}
RIS
TY - UNPB
TI - Finite groups that are the product of every pair of non-conjugate maximal subgroups are soluble
AU - Richie Sater
PY - 2026
DA - 2026/08/27
PB - arXiv
DO - 10.48550/arxiv.2608.02970
UR - https://arxiv.org/abs/2608.02970
AB - We prove that every finite group that is the product of every pair of its non-conjugate maximal subgroups is soluble, answering Problem 10.34 of the Kourovka Notebook. The almost-simple case was proved by Tikhonenko and Tyutyanov. We treat the remaining minimal-counterexample branch, where the unique minimal normal subgroup has the form S^k, with S nonabelian simple and k >= 2. Two automorphism-stable coordinate subgroup classes satisfying a maximal-supplement criterion produce non-conjugate maximal supplements. If the factorization hypothesis held, their product would impose a p-adic divisibility requirement growing linearly with k, while the quotient contributes only coordinate outer automorphisms and a factor dividing k!. A fixed valuation gap therefore excludes every k >= 2 at once. Suitable subgroup classes are constructed uniformly across the infinite families of finite simple groups using parabolic, torus, and primitive-prime-divisor arguments; stable flag parabolics handle graph fusion, while GAP certificates cover designated finite and sporadic cases. The resulting all-k obstruction provides a reusable mechanism for eliminating direct-power socles in finite-group factorization problems.
ER -