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# The maximal size of a minimal generating set

Mar 2023

A generating set for a finite group $G$ is said to be minimal if no propersubset generates $G$, and $m(G)$ denotes the maximal size of a minimalgenerating set for $G$. We prove a conjecture of Lucchini, Moscatiello andSpiga 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, wefirst bound $m(G)$ for all almost simple groups of Lie type (until now, nonontrivial bounds were known except for groups of rank $1$ or $2$). Inparticular, we prove that there exist $a,b > 0$ such that any finite simplegroup $G$ 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 $\omega(f)$ denotesthe number of distinct prime divisors of $f$. In the process, we confirm aconjecture of Gill and Liebeck that there exist $a,b > 0$ such that a minimalbase for a faithful primitive action of an almost simple group of Lie type ofrank $r$ over $\mathbb{F}_{p^f}$ has size at most $ar^b + \omega(f)$.

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