PreprintarXiv licencearXiv
Finite Soluble Groups Need Not Admit Increasing Unrefinable Subgroup Chains
Abstract
In correspondence beginning on August 8, 2026, V. S. Monakhov and I. L. Sokhor communicated the following question to the author: does every finite group have an unrefinable chain of subgroups whose successive indices are nondecreasing? We answer this question in the negative, even among soluble groups. For every odd prime p, we construct a soluble matrix group of degree five over the field with p elements. The group has order 64 times the eighth power of p and has no such chain. These counterexamples form an infinite family whose smallest member, obtained when p equals three, has order 419904. The construction is based on an example due to Kohler, and the obstruction follows from three elementary calculations of maximal subgroup indices.
Related
- The Conjugacy Problem for Recursively Presented Virtually Soluble Groups of Finite Prüfer Rank
- Finite groups that are the product of every pair of non-conjugate maximal subgroups are soluble
- M-subgroups and normal-subgroup lattices of finite groups
- The congruence subgroup problem for branch groups
- Supersoluble groups and the probability of generating a supersoluble subgroup
BibTeX
@misc{sater2026finite,
title = {{Finite Soluble Groups Need Not Admit Increasing Unrefinable Subgroup Chains}},
author = {Richie Sater},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26228},
eprint = {2609.26228},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26228},
}
RIS
TY - UNPB
TI - Finite Soluble Groups Need Not Admit Increasing Unrefinable Subgroup Chains
AU - Richie Sater
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.26228
UR - https://arxiv.org/abs/2609.26228
AB - In correspondence beginning on August 8, 2026, V. S. Monakhov and I. L. Sokhor communicated the following question to the author: does every finite group have an unrefinable chain of subgroups whose successive indices are nondecreasing? We answer this question in the negative, even among soluble groups. For every odd prime p, we construct a soluble matrix group of degree five over the field with p elements. The group has order 64 times the eighth power of p and has no such chain. These counterexamples form an infinite family whose smallest member, obtained when p equals three, has order 419904. The construction is based on an example due to Kohler, and the obstruction follows from three elementary calculations of maximal subgroup indices.
ER -