Prime power coverings of groups
Michael Giudici, Luke Morgan, Cheryl E. Praeger

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 with normal subgroup , a subgroup of is an -prime-power-covering subgroup if meets every -conjugacy-class of elements of of prime power order. It is conjectured that is bounded by some function of , 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 on the set of right cosets of in is innately transitive. This includes the case where is a maximal subgroup of . 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.
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Taxonomy
TopicsFinite Group Theory Research · Rings, Modules, and Algebras · Algebraic Geometry and Number Theory
