Conjugacy class sizes in arithmetic progression

2020/03/09 by Bianchi, Mariagrazia, Praeger, Cheryl E., Glasby, S. P.
#20D-60 #20E-45 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2003.03906

Abstract

Let \rm cs(G) denote the set of conjugacy class sizes of a group G, and let \rm cs^*(G)=\rm cs(G)∖\1\ be the sizes of non-central classes. We prove three results. We classify all finite groups G with \rm cs(G)=\a, a+d, … ,a+rd\ an arithmetic progression with r\geqslant 2. (We show that \rm cs(G)=\1,2,3\.) Our most substantial result classifies all G with \rm cs^*(G)=\2,4,6\. Finally, we classify all groups G whose largest two non-central conjugacy class sizes are coprime. (Here it is not obvious but it is true that \rm cs^*(G) has two elements, and so is an arithmetic progression.)

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