Finite 2-groups with odd number of conjugacy classes

TL;DR

This paper investigates finite 2-groups with an odd number of real conjugacy classes, establishing finiteness for certain counts and constructing infinite families for others using advanced algebraic techniques.

Contribution

It proves finiteness of 2-groups with odd real conjugacy classes for counts less than 24 and constructs infinite examples for 25, using pro-p and Kneser classification methods.

Findings

Finitely many 2-groups with odd real conjugacy classes for counts less than 24.

Existence of infinitely many 2-groups with exactly 25 real conjugacy classes.

Abstract

In this paper we consider finite 2-groups with odd number of real conjugacy classes. On one hand we show that if $k$ is an odd natural number less than 24, then there are only finitely many finite 2-groups with exactly $k$ real conjugacy classes. On the other hand we construct infinitely many finite 2-groups with exactly 25 real conjugacy classes. Both resuls are proven using pro-$p$ techniques and, in particular, we use the Kneser classification of semi-simple $p$-adic algebraic groups.