Coprime invariable generation and minimal-exponent groups

TL;DR

This paper introduces the concept of coprime invariable generation in finite groups, showing that such groups have minimal generating sets and zero presentation rank, with all finite simple groups being coprimely-invariably generated.

Contribution

It establishes new properties of coprimely-invariably generated groups, including bounds on generators and presentation rank, and proves all finite simple groups possess this property.

Findings

Coprimely-invariably generated groups can be generated with at most three elements.

Finite simple groups are all coprimely-invariably generated.

Groups with no proper subgroup sharing the same exponent have zero presentation rank.

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 $\prod_{i=1}^u k_{\pi_i}(S) \leq \frac{|S|}{2| Out S|}.$