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

Abstract: Let $N$ be normal subgroup of a finite group $G$, $p$ be a prime, $P$ be a Sylow $p$-subgroup of $G$ and $θ$ 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 $θ$ 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 $θ$.

• The paper establishes an A-equivariant bijection between p'-degree irreducible characters of G and those in the normalizer subgroup over a fixed irreducible character.
• It adapts classical techniques like Gallagher’s theorem and the Glauberman correspondence to address automorphism actions without full reliance on inductive McKay reductions.
• The results offer new theoretical insights and computational methods for managing equivariant character correspondences in finite group representation theory.

Equivariant McKay Correspondence for $p$-Solvable Groups via the Okuyama-Wajima Argument

Introduction and Context

This work addresses a refined version of the McKay conjecture, extending its scope to include the action of group automorphisms and focusing on the case where $G/N$ is $p$-solvable. Specifically, the main result establishes the existence of an $A$-equivariant bijection between irreducible characters of $G$ lying over a $P$-invariant irreducible character $\theta$ of a normal subgroup $N$, of degree prime to $p$, and those of the corresponding normalizer subgroup, with $A$ a finite group acting on $G$ stabilizing $N$ and $P$. This is shown by reworking and generalizing classical arguments, notably those of Gallagher and Okuyama-Wajima, while avoiding the deeper machinery involving endo-$p$-permutation modules and the inductive McKay condition.

Main Results

Theorem A: $A$-Equivariant McKay Bijection for $p$-Solvable Extensions

Let $N \trianglelefteq G$ and $A$ a finite group acting on $G$ by automorphisms, stabilizing $N$ and a Sylow $p$-subgroup $P$ of $G$, and let $G/N$ be $p$-solvable. For a $P$-invariant irreducible character $\theta$ of $N$, there exists an $A$-equivariant bijection between irreducible characters of $G$ of degree prime to $p$ lying above $\theta$ and such characters in $\norm{G}{P}N$ lying over $\theta$ (2512.13406).

This result simultaneously generalizes and strengthens previous results. Unlike existing approaches, which relied on the full reduction to simple groups and the inductive McKay condition (e.g., [IMN07], [Ros23]), the proof here uses foundational tools: the Okuyama-Wajima argument, the Glauberman correspondence, and a novel equivariant extension of Gallagher’s classical counting result.

Theorem B: Equivariant Counting of Irreducible Characters

If $N$ and $G$ are normal in $A$ and $\theta \in \irr N$ is $A$-invariant, then the number of $A$-invariant irreducible characters of $G$ lying over $\theta$ equals the number of conjugacy classes of $G/N$ that are $\theta$-good in $A$. The notion of $\theta$-goodness is central, as it captures elements with suitable extension properties for $\theta$, generalizing the classical setting to the equivariant context.

The Okuyama-Wajima Technical Framework

Central to the reduction is the deployment of Okuyama-Wajima's method: for a $p$-group $Q$ acting on a $p'$-group $K$, the character theory relating $K$ and its fixed points under $Q$ is mediated by the Glauberman correspondence. This interplay is crucial in handling character extensions and transfer under automorphism actions. The paper proves key consequences for equivariant settings, for instance:

• The number of irreducible, $A$-invariant (and $p'$-degree) characters of $G$ covering a product $\theta \times \lambda$ (where $\lambda$ is an $A$-invariant linear character of a central $p$-subgroup) matches the analogous count for the fixed-point subgroup, equipped with the Glauberman correspondent $\theta^*$.

This mechanism is then upscaled to $p$-solvable group extensions via a careful induction strategy, iterating upon normal subgroups and combining counting results with character triple isomorphisms.

Technical Framework and Methods

The proof architecture combines several deep ingredients:

• Strong Isomorphisms of Character Triples: Using Isaacs's and Navarro’s formalism, character triples $(A,N,\theta)$ are replaced by strongly isomorphic triples with central normal subgroups, simplifying extension and invariance questions.
• Gallagher’s Theorem and Extensions: The number of irreducible characters in a group lying over a fixed irreducible character of a normal subgroup is equated to the number of "good" conjugacy classes, with "goodness" now suitably generalized to equivariant and, in places, Glauberman settings.
• Inductive Reduction: The proof reduces along chief series for $G/N$, relying on $p$-solvability to iterate the argument via normal sections that are either $p$ or $p'$-groups, always maintaining control over automorphism actions.
• Detailed Clifford and Glauberman Theory: Key bijections, invariance properties, and counting results are established using generalized Clifford theory, complemented by the Glauberman correspondence’s equivariance under automorphisms.

Implications

Theoretical Significance

The results provide an independent, automorphism-compatible proof of the relative McKay conjecture for $p$-solvable groups, achieving equivariance without resorting to the full strength of the inductive McKay reduction theorem. This approach exposes finer structure in the interaction of group automorphisms, character invariance, and bijections underpinning the McKay philosophy.

Notably, the work provides an explicit and elementary toolkit for further exploration of global-to-local correspondences in character theory, especially in the equivariant and relative settings, and opens new ground for understanding the role of extensions and "good" classes in block-theoretic conjectures.

Practical Consequences and Potential Extensions

• Automorphism-Compatible Representation Theory: The technical results can be employed directly in problems where equivariance under additional group actions is necessary, for instance in blockwise equivariant correspondences in modular representation theory.
• Broader Application to Alperin-Type and Glauberman-Type Conjectures: The techniques and concepts developed can inform work on related equi-character correspondences, such as blockwise Alperin and Isaacs-Navarro conjectures, particularly where canonical bijections must be maintained with respect to larger automorphism groups.
• Computational Character-Theoretic Algorithms: As the counting methods are explicit and combinatorial, they may guide the development of structural algorithms for character analysis in computational group theory, especially for $p$-solvable groups.

Conclusion

This paper demonstrates that the relative McKay conjecture, even in the presence of group automorphisms, can be established for $p$-solvable groups by direct technical means, centering the Okuyama-Wajima argument and a new, fully equivariant Gallagher-type counting theorem. This circumvents reliance on the inductive McKay condition, elucidates equivariant character behavior, and streamlines proofs for the $p$-solvable case, while offering methodological advances of independent interest. The results underscore robust connections between automorphism actions, character extensions, and local-global correspondences in finite group representation theory (2512.13406).