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 $m(G) \leq a \cdot \delta(G)^b$, for $\delta(G) = \sum_{\text{$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 $r + \omega(f) \leq m(G) \leq a(r + \omega(f))^b$, where $\omega(f)$ 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 $ar^b + \omega(f)$.