Restrictions on sets of conjugacy class sizes in arithmetic progressions

2020/09/11 by A. R. Camina, Camina, Alan R., Rachel D. Camina +1
Computer Science · Mathematics · #20E45 #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.2009.05355

openalex publication_date 2020/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We continue the investigation, that began in [3] and [4], into finite groups whose set of nontrivial conjugacy class sizes form an arithmetic progression. Let G be a finite group and denote the set of conjugacy class sizes of G by \rm cs(G). Finite groups satisfying \rm cs(G) = \1,2,4,6\ and \1,2,4,6,8\ are classified in [4] and [3], respectively, we demonstrate these examples are rather special by proving the following. There exists a finite group G such that \rm cs(G) = \1, 2^α, 2^α+1, 2^α3 \ if and only if α=1. Furthermore, there exists a finite group G such that \rm cs(G) = \1, 2^α, 2^α+1, 2^α3, 2^α+2\ and α is odd if and only if α=1.

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