Groups with some arithmetic conditions on real class sizes

TL;DR

This paper investigates finite groups where real conjugacy class sizes are not divisible by 4, proving such groups are solvable and analyzing their structure, thus extending previous results in group theory.

Contribution

It establishes a new solvability criterion based on the divisibility properties of real conjugacy class sizes in finite groups.

Findings

Groups with no real conjugacy class size divisible by 4 are solvable

Detailed structural analysis of such groups

Generalizes previous results in the literature

Abstract

Let G be a finite group. An element x in G is a real element if x is conjugate to its inverse in G. For x in G, the conjugacy class x^G is said to be a real conjugacy class if every element of x^G is real. We show that if 4 divides no real conjugacy class size of a finite group G, then G is solvable. We also study the structure of such groups in detail. This generalizes several results in the literature.