2-Parts of real class sizes

TL;DR

This paper proves that finite groups with certain uniform real class size properties are solvable and have a specific 2-length, confirming a conjecture linking real class sizes to group solvability.

Contribution

It establishes that groups with non-central real class sizes sharing the same 2-part are solvable with 2-length one, confirming a conjecture by Navarro, Sanus, and Tiep.

Findings

Groups with uniform non-central real class size 2-parts are solvable.

Such groups have 2-length one.

Finite groups with only two real class sizes are solvable.

Abstract

We investigate the structure of finite groups whose non-central real class sizes have the same $2$-part. In particular, we prove that such groups are solvable and have $2$-length one. As a consequence, we show that a finite group is solvable if it has two real class sizes. This confirms a conjecture due to G. Navarro, L. Sanus and P. Tiep.