Restrictions on sets of conjugacy class sizes in arithmetic progressions

TL;DR

This paper investigates the structure of finite groups whose nontrivial conjugacy class sizes form specific arithmetic progressions, revealing restrictions and classifications for certain sets of class sizes.

Contribution

It characterizes when finite groups can have conjugacy class sizes forming particular arithmetic progressions involving powers of two and multiples of three.

Findings

Existence of groups with class sizes {1, 2^α, 2^{α+1}, 2^α 3} only when α=1.

Existence of groups with class sizes {1, 2^α, 2^{α+1}, 2^α 3, 2^{α+2}} only when α=1 and α is odd.

Classifications show such class size sets are highly restricted and special.

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