The McKay conjecture with group automorphisms and the Okuyama-Wajima argument

TL;DR

This paper proves an A-equivariant McKay bijection for finite groups with certain automorphism actions, extending previous results and employing the Okuyama-Wajima argument to handle characters over Glauberman correspondents.

Contribution

It introduces an A-equivariant McKay bijection for groups with automorphisms, independent of Rossi's recent work, and generalizes Gallagher's classical result.

Findings

Establishes an A-equivariant bijection between specific irreducible characters.

Extends the McKay conjecture framework to include group automorphisms.

Utilizes the Okuyama-Wajima argument for characters over Glauberman correspondents.

Abstract

Let $N$ be normal subgroup of a finite group $G$, $p$ be a prime, $P$ be a Sylow $p$-subgroup of $G$ and $\theta$ be a $P$-invariant irreducible character of $N$. Suppose that $G/N$ is a $p$-solvable group. In this note we show that, whenever a finite group $A$ acts on $G$ stabilizing $P$, there exists an $A$-equivariant McKay bijection between irreducible characters lying over $\theta$ of degree prime to $p$ of $G$ and $\textbf{N}_G(P)$. This is a consequence of a recent result of D. Rossi. Our approach here is independent from Rossi's and follows the original idea of the proof of the McKay conjecture for $p$-solvable groups. In particular, we rely on the so-called Okuyama-Wajima argument to deal with characters above Glauberman correspondents. For this purpose, we generalize a classical result of P. X. Gallagher on the number of irreducible characters of $G$ lying over $\theta$.