Total closure for permutation actions of finite nonabelian simple groups

TL;DR

This paper determines the minimal number of Cartesian factors needed for finite simple groups to be totally closed under group actions, revealing exact values for alternating groups and bounds for classical groups.

Contribution

It provides the first comprehensive bounds and exact values for the closure number $k(G)$ for various classes of finite simple groups, including alternating and classical groups.

Findings

$k(A_n)=n-1$ for alternating groups

$k(G) ext{ is at most }7$ for most simple groups

$k(G) ext{ is at most } n+2$ for classical groups with dimension } n$

Abstract

For a positive integer $k$, a group $G$ is said to be totally $k$-closed if for each set $\Omega$ upon which $G$ acts faithfully, $G$ is the largest subgroup of $\mathrm{Sym}(\Omega)$ that leaves invariant each of the $G$-orbits in the induced action on $\Omega\times\cdots\times \Omega=\Omega^k$. Each finite group $G$ is totally $|G|$-closed, and $k(G)$ denotes the least integer $k$ such that $G$ is totally $k$-closed. We address the question of determining the closure number $k(G)$ for finite simple groups $G$. Prior to our work it was known that $k(G)=2$ for cyclic groups of prime order and for precisely six of the sporadic simple groups, and that $k(G)\geq3$ for all other finite simple groups. We determine the value for the alternating groups, namely $k(A_n)=n-1$. In addition, for all simple groups $G$, other than alternating groups and classical groups, we show that $k(G)\leq 7$. Finally, if $G$ is a finite simple classical group with natural module of dimension $n$, we show that $k(G)\leq n+2$ if $n \ge 14$, and $k(G) \le \lfloor n/3 + 12 \rfloor$ otherwise, with smaller bounds achieved by certain families of groups. This is achieved by determining a uniform upper bound (depending on $n$ and the type of $G$) on the base sizes of the primitive actions of $G$, based on known bounds for specific actions. We pose several open problems aimed at completing the determination of the closure numbers for finite simple groups.