The maximal size of a minimal generating set

Abstract

A generating set for a finite group $G$ is said to be minimal if no proper subset generates $G$, and $m(G)$ denotes the maximal size of a minimal generating set for $G$. We prove a conjecture of Lucchini, Moscatiello and Spiga by showing that there exist $a,b>0$ such that any finite group $G$ satisfies $m(G)\le a\cdot \hat{I}´(G{)}^b$, for $δ(G) = \sum_{\text{$p$ prime}} m(G_p)$ where $G_p$ is a Sylow $p$-subgroup of $G$. To do this, we first bound $m(G)$ for all almost simple groups of Lie type (until now, no nontrivial bounds were known except for groups of rank $1$ or $2$). In particular, we prove that there exist $a,b>0$ such that any finite simple group $G$ of Lie type of rank $r$ over the field ${\mathbb{F}}_{p^f}$ satisfies $r+\ddot{I}‰(f)\le m(G)\le a(r+\ddot{I}‰(f){)}^b$, where $\ddot{I}‰(f)$ denotes the number of distinct prime divisors of $f$. In the process, we confirm a conjecture of Gill and Liebeck that there exist $a,b>0$ such that a minimal base for a faithful primitive action of an almost simple group of Lie type of rank $r$ over ${\mathbb{F}}_{p^f}$ has size at most $a{r}^b+\ddot{I}‰(f)$.

10 pages; to appear in Forum of Mathematics, Sigma