Prime power coverings of groups

TL;DR

This paper investigates prime power coverings in finite groups, proving a conjecture under specific transitivity conditions and introducing a new bound on simple groups based on element classes, with implications for algebraic number theory.

Contribution

It proves a conjecture about prime power coverings in finite groups under certain conditions and introduces a new bound on simple groups based on element class counts.

Findings

Proved the conjecture for innately transitive group actions.

Established a new bound on nonabelian simple groups related to prime power element classes.

Connected group-theoretic results to number theoretic implications for algebraic number fields.

Abstract

For a finite group $A$ with normal subgroup $G$, a subgroup $U$ of $G$ is an $A$-prime-power-covering subgroup if $U$ meets every $A$-conjugacy-class of elements of $G$ of prime power order. It is conjectured that $|G:U|$ is bounded by some function of $|A:G|$, and this conjecture has number theoretic implications for relative Brauer groups of algebraic number fields. We prove the conjecture in the case that the action of $G$ on the set of right cosets of $U$ in $G$ is innately transitive. This includes the case where $U$ is a maximal subgroup of $G$. The proof uses a new bound on the order of a nonabelian finite simple group in terms of its number of classes of elements of prime power order, which in turn depends on the Classification of the Finite Simple Groups.