有限可解群不一定具有递增的不可加细子群链
Finite Soluble Groups Need Not Admit Increasing Unrefinable Subgroup Chains
AI总结：
本文否定回答了Monakhov和Sokhor提出的问题：并非每个有限群（即使可解）都有递增指标的不可加细子群链，并通过构造无限族反例（最小阶419904）证明。
AI中文摘要：
在2026年8月8日开始的通信中，V. S. Monakhov和I. L. Sokhor向作者提出了以下问题：每个有限群是否都具有一条不可加细的子群链，使得其连续指标是非递减的？我们对该问题给出了否定回答，即使在可解群中也是如此。对于每个奇素数p，我们构造了一个在含p个元素的域上的五次可解矩阵群。该群的阶为64乘以p的八次方，并且不具有这样的链。这些反例构成一个无限族，其中最小的成员在p等于3时取得，其阶为419904。该构造基于Kohler的一个例子，而障碍则来自对极大子群指标的三个初等计算。
英文摘要：
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.
