Independent sets of generators of prime power order

TL;DR

This paper introduces the concept of prime-power-independent generating sets in finite groups, characterizes groups where all such sets have the same size, and proves these groups are solvable, leading to a full classification.

Contribution

It defines prime-power independence in groups, proves that groups with uniform prime-power-independent generating set sizes are solvable, and provides a complete classification of these groups.

Findings

Groups with uniform prime-power-independent generating sets are solvable

Complete classification of $_{pp}$-groups achieved

Connection established with recent results of Krempa and Stocka

Abstract

A subset $X$ of a finite group $G$ is said to be prime-power-independent if each element in $X$ has prime power order and there is no proper subset $Y$ of $X$ with $\langle Y, \Phi(G)\rangle = \langle X, \Phi(G)\rangle$, where $\Phi(G)$ is the Frattini subgroup of $G$. A group $G$ is $\mathcal{B}_{pp}$ if all prime-power-independent generating sets for $G$ have the same cardinality. We prove that, if $G$ is $\mathcal{B}_{pp}$, then $G$ is solvable. Pivoting on some recent results of Krempa and Stocka, this yields a complete classification of $\mathcal{B}_{pp}$-groups.