A lower bound for the number of odd-degree representations of a finite group

TL;DR

This paper establishes bounds on the number of odd-degree irreducible representations of a finite group, linking it to the structure of its Sylow 2-subgroup and leveraging recent progress on the McKay conjecture.

Contribution

It provides new asymptotic and explicit bounds for these representations based on the abelianization of the Sylow 2-subgroup, combining recent conjecture proofs and class number bounds.

Findings

Derived bounds relate odd-degree representations to Sylow 2-subgroup structure.

Utilized recent proof of the McKay conjecture for prime 2.

Proved lower bounds for class numbers of semidirect products.

Abstract

Let $G$ be a finite group and $P$ a Sylow $2$-subgroup of $G$. We obtain both asymptotic and explicit bounds for the number of odd-degree irreducible complex representations of $G$ in terms of the size of the abelianization of $P$. To do so, we, on one hand, make use of the recent proof of the McKay conjecture for the prime 2 by Malle and Sp\"{a}th, and, on the other hand, prove lower bounds for the class number of the semidirect product of an odd-order group acting on an abelian $2$-group.