Kourovka Notebook Problem 21.55

Definitions and mathematical statement

Fix a prime power $q=p^a$, with $p$ prime and $a\ge1$, and let $\mathbb F_q$ be the field with $q$ elements. A finite group $H$ is $p$-soluble if it has a finite normal series whose factors are either $p$-groups or groups of order prime to $p$. The $p$-length $\ell_p(H)$ is the minimum number of nontrivial $p$-group factors among such series. The factors of order prime to $p$ do not contribute to this count. In particular a group of order prime to $p$ has $p$-length zero.

For each positive integer $n$, define \[ m_n(q)=\max\{\ell_p(H):H\le\operatorname{GL}_n(\mathbb F_q), \ H\text{ is }p\text{-soluble}\}. \] The maximum exists because the ambient general linear group is finite. Subgroups need not act irreducibly, and are not required to be soluble at primes other than the specified $p$.

The question is whether, for every fixed $q$, \[ \lim_{n\to\infty}\frac{m_n(q)}{\log_2 n}=1. \] Equivalently, for every $\varepsilon>0$ there should be $N(q,\varepsilon)$ such that for all $n\ge N(q,\varepsilon)$ the ratio differs from one by less than $\varepsilon$.

The field is fixed before the dimension tends to infinity; uniformity as $q$ varies is not requested. The logarithm is specifically base two, so its choice fixes the leading constant. Both an asymptotic upper bound for all eligible subgroups and a matching lower construction are needed for the asserted limit.

Short English statement

For a fixed finite field of characteristic $p$, is the maximal $p$-length of a $p$-soluble linear subgroup asymptotic to the base-two logarithm of the dimension?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2564 (KOU-21.55), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, new problems, Problem 21.55. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.