Finite Soluble Groups Need Not Admit Increasing Unrefinable Subgroup Chains

Richie Sater

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arXiv · 2609.26228

Finite Soluble Groups Need Not Admit Increasing Unrefinable Subgroup Chains

Richie Sater

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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.

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BibTeX RIS

Richie Sater. 2026-08-11. Finite Soluble Groups Need Not Admit Increasing Unrefinable Subgroup Chains. https://arxiv.org/abs/2609.26228

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