Invariable generation with elements of coprime prime-power order

TL;DR

This paper investigates a special type of finite group generation involving elements with pairwise coprime prime-power orders, establishing relationships and equivalences between these properties, especially in soluble groups.

Contribution

It introduces and analyzes coprime prime-power invariable generation, proving its relation to coprime invariable generation and their equivalence in soluble groups.

Findings

Prime-power coprime invariable generation is stronger than coprime invariable generation.

In finite soluble groups, both properties are equivalent.

The paper provides proofs relating these generation properties.

Abstract

A finite group $G$ is coprimely-invariably generated if there exists a set of generators $\{g_1, \ldots, g_d\}$ of $G$ with the property that the orders $|g_1|, \ldots, |g_d|$ are pairwise coprime and that for all $x_1, \ldots, x_d \in G$ the set $\{g_1^{x_1}, \ldots, g_d^{x_d}\}$ generates $G$. In the particular case when $|g_1|, \ldots, |g_d|$ can be chosen to be prime-powers we say that $G$ is prime-power coprimely-invariably generated. We will discuss these properties, proving also that the second one is stronger than the first, but that in the particular case of finite soluble groups they are equivalent.