Coprime invariable generation and minimal-exponent groups

Abstract

A finite group $G$ is \emph{coprimely-invariably generated} if there exists a set of generators $\{g_1,...,g_u\}$ of $G$ with the property that the orders $|g_1|,...,|g_u|$ are pairwise coprime and that for all $x_1,...,x_u\in G$ the set $\{g_1^{x_1},...,g_u^{x_u}\}$ generates $G$. We show that if $G$ is coprimely-invariably generated, then $G$ can be generated with three elements, or two if $G$ is soluble, and that $G$ has zero presentation rank. As a corollary, we show that if $G$ is any finite group such that no proper subgroup has the same exponent as $G$, then $G$ has zero presentation rank. Furthermore, we show that every finite simple group is coprimely-invariably generated. Along the way, we show that for each finite simple group $S$, and for each partition ${\pi }_1,...,{\pi }_u$ of the primes dividing $|S|$, the product of the number $k_{{\pi }_i}(S)$ of conjugacy classes of ${\pi }_i$-elements satisfies ${\prod }_{i=1}^u{k}_{{\pi }_i}(S)\le \frac{|S|}{2|OutS|}.$