A Negative Answer to a Question on Monotone Maximal Chains in Finite Groups

We give a negative answer, even among soluble groups, to the question whether every finite group has an unrefinable subgroup chain whose successive indices are nondecreasing. For every odd prime p, we construct a soluble matrix group Gp ≤ GL5(p) of order 26p8 with no such chain. The smallest member of this infinite family, G3, has order 419904; no claim of global minimality is made. This deposit contains the five-page mathematical note, a standalone GAP certificate for G3, successful verification logs for GAP 4.11.1 and 4.16.0, a source-and-verification archive, and SHA-256 checksums. The manuscript, documentation, and captured textual results are licensed CC BY 4.0; source code is licensed MIT as specified in the included LICENSE file.