Subgroups of simple groups are as diverse as possible

TL;DR

This paper establishes bounds on the number and diversity of subgroups in finite simple groups, showing they are as diverse as possible except in some bounded cases, with implications for understanding their subgroup structure.

Contribution

It provides asymptotic bounds on the number and types of subgroups in various classes of finite simple groups, revealing their maximal diversity in most cases.

Findings

Bounds on the number of isomorphism types of subgroups in simple groups.

Subgroups of most finite simple groups are highly diverse.

Finite simple groups have maximal subgroup diversity except in bounded cases.

Abstract

For a finite group $G$, let $\sigma(G)$ be the number of subgroups of $G$ and $\sigma_\iota(G)$ the number of isomorphism types of subgroups of $G$. Let $L=L_r(p^e)$ denote a simple group of Lie type, rank $r$, over a field of order $p^e$ and characteristic $p$. If $r\neq 1$, $L\not\cong {^2 B_2}(2^{1+2m})$, then there are constants $c,d$, dependent on the Lie type, such that as $re$ grows $$p^{(c-o(1))r^4e^2}\leq\sigma_{\iota}(L_r(p^e))\leq\sigma(L_r(p^e)) \leq p^{(d+o(1))r^4e^2}.$$ For type $A$, $c=d=1/64$. For other classical groups $1/64\leq c\leq d\leq 1/4$. For exceptional and twisted groups $1/2^{100}\leq c\leq d\leq 1/4$. Furthermore, $$2^{(1/36-o(1))k^2)}\leq\sigma_{\iota}(\mathrm{Alt}_k)\leq \sigma(\mathrm{Alt}_k)\leq 24^{(1/6+o(1))k^2}.$$ For abelian and sporadic simple groups $G$, $\sigma_{\iota}(G),\sigma(G)\in O(1)$. In general these bounds are best possible amongst groups of the same orders. Thus with the exception of finite simple groups with bounded ranks and field degrees, the subgroups of finite simple groups are as diverse as possible.