Covering the set of p-elements in finite groups by proper subgroups

Maróti, Attila and Martínez, Juan and Moretó, Alexander (2025) Covering the set of p-elements in finite groups by proper subgroups. JOURNAL OF COMBINATORIAL THEORY SERIES A, 210. ISSN 0097-3165

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Abstract

Let p be a prime and let G be a finite group which is generated by the set Gp of its p-elements. We show that if G is solvable and not a p-group, then the minimal number σp(G) of proper subgroups of G whose union contains Gp is equal to 1 less than the minimal number of proper subgroups of G whose union is G. For p-solvable groups G, we always have σp(G)≥p+1. We study the case of alternating and symmetric groups G in detail. © 2024 The Author(s)

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