Real class sizes

TL;DR

This paper investigates the structure of finite groups through the sizes of their real conjugacy classes, establishing conditions under which the group is solvable based on prime graph connectivity and class size properties.

Contribution

It introduces new criteria linking real class size arithmetic conditions to the solvability of finite groups, expanding understanding of group structure analysis.

Findings

Disconnected prime graph implies group solvability

Uniform 2-part sizes of real classes lead to solvability

Conditions on Sylow 2-subgroups influence group structure

Abstract

In this paper, we study the structures of finite groups using some arithmetic conditions on the sizes of real conjugacy classes. We prove that a finite group is solvable if the prime graph on the real class sizes of the group is disconnected. Moreover, we show that if the sizes of all non-central real conjugacy classes of a finite group $G$ have the same $2$-part and the Sylow $2$-subgroup of $G$ satisfies certain condition, then $G$ is solvable.