On A-Groups with the Same Index Set as a Nilpotent Group

2025/06/18 by Wei Zhou, Zhou, Wei, Ilya Gorshkov +1
Engineering · Mathematics · #20D15 #20D60 #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Rings, Modules, and Algebras #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2506.15250

openalex publication_date 2025/06/18 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28

Abstract

Let G be a finite group and N(G) be the set of conjugacy class sizes of G. For a prime p, let |G||_p be the highest p-power dividing some element of N(G). and define |G|| = Π_{p∈ π(G)}|G||_p. G is said to be an A-group if all its Sylow subgroups are abelian. We prove that if G is an A-group such that N(G) contains |G||_p for every p∈ π(G) as well as |G||, then G must be abelian. This result gives a positive answer to a question posed by Camina and Camina in 2006.

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