Restrictions on sets of conjugacy class sizes in arithmetic progressions

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 $\mathrm{cs}(G)$. Finite groups satisfying $\mathrm{cs}(G)=\{1,2,4,6\}$ 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 $\mathrm{cs}(G)=\{1,2^{\alpha },2^{\alpha +1},2^{\alpha }3\}$ if and only if $\alpha =1$. Furthermore, there exists a finite group $G$ such that $\mathrm{cs}(G)=\{1,2^{\alpha },2^{\alpha +1},2^{\alpha }3,2^{\alpha +2}\}$ and $\alpha$ is odd if and only if $\alpha =1$.