Okuyama–Wajima Argument in Finite Group Theory

• The Okuyama–Wajima argument is a finite group theory framework that reduces complex character extension problems to abelian p-quotients using Glauberman correspondences.
• It employs an equivariant refinement of Gallagher’s theorem to precisely count and match irreducible characters in the presence of automorphisms.
• This method underpins automorphism-equivariant McKay bijections for p-solvable groups, providing systematic control over the enumeration of irreducible characters.

The Okuyama–Wajima argument is a pivotal technique in finite group theory and character theory with applications in the proof of the McKay conjecture for $p$-solvable groups, including cases involving group automorphisms. Originating from a 1980 theorem of Okuyama and Wajima, its central insight is that extension problems for irreducible characters of a normal $p'$-subgroup, tightly linked to the behavior of Glauberman correspondents and the control exerted by Gallagher counting, become tractable by a reduction to abelian $p$-quotients. The framework enables explicit matching of characters in equivariant correspondences, particularly in producing $A$-equivariant bijections in the presence of automorphisms. Recent elaborations leverage these combinatorial and representation-theoretic features to provide self-contained proofs of equivariant versions of the McKay conjecture for finite $p$-solvable groups (Maltempo et al., 15 Dec 2025).

1. Technical Framework and Preliminaries

Let $G$ be a finite group, $p$ a prime, $N\unlhd G$ a normal subgroup, $P\in\syl pG$ a Sylow $p$-subgroup, $\theta\in\Irr(N)$ a $P$-invariant irreducible character, and $A$ a finite group acting on $G$ stabilizing $P$. The set of irreducible characters of $G$ above $\theta$ is denoted $\Irr(G\mid\theta)$, with those of degree prime to $p$ as $\Irr_{p'}(G\mid\theta)$. Fundamental to the argument is the Glauberman correspondence: when a $p$-group $Q\le\Aut(K)$ acts on a finite group $K$ with $(|K|,p)=1$, one has a canonical bijection

${}^*\;:\;\Irr_{Q}(K)\;\longrightarrow\;\Irr(\Cent_K(Q)),$

where $\theta^*$ is the Glauberman correspondent of a $Q$-fixed $\theta\in\Irr(K)$.

This landscape sets up the study of how irreducible characters lying over a given $\theta$ can be controlled and matched, particularly when targeted restrictions and extensions interplay with automorphism groups.

2. Gallagher’s Theorem and Equivariant Counting

Classical Gallagher’s theorem establishes that for $N\unlhd A$ and $\theta\in\Irr(N)$ $A$-invariant, the number $|\Irr(A\mid\theta)|$ equals the number of $A/N$-conjugacy classes with $\theta$-good representatives. The Okuyama–Wajima argument deploys an equivariant refinement (Theorem A (Maltempo et al., 15 Dec 2025)): for $N\unlhd G\unlhd A$ with $A$ acting and $\theta$ $A$-invariant, the cardinality of $A$-fixed characters of $G$ over $\theta$ equals the number of $G/N$-conjugacy classes all of whose elements are $\theta$-good in $A$. Navarro’s function space isomorphism (Theorem B) recasts this as a linear-algebraic statement: the restriction map from class functions on $A$ vanishing off $G$ and affording $\theta$ on $N$ gives a vector space with natural bases indexed by $A$-fixed irreducibles and by classes of $\theta$-good elements.

This combination provides precise enumerative control, ensuring bijectivity in subsequent character correspondences and supporting the main reduction steps in the Okuyama–Wajima argument.

3. The Okuyama–Wajima Extension Theorem

The Okuyama–Wajima Extension Theorem asserts: let $A$ act on $K$ with $(|K|,p)=1$, $Q\le A$ normalizing $K$ a $p$-subgroup, $B=N_A(Q)$, $C=\Cent_K(Q)$. For $Q$-fixed $\theta\in\Irr(K)$ with Glauberman correspondent $\theta^*\in\Irr(C)$, and $C\le U\le B$ with $U/C$ abelian,

$$\theta \text{ extends to } K U \quad \Longleftrightarrow \quad \theta^* \text{ extends to } U.$$

This reduces the extension problem for characters across semidirect products containing a $p$-group quotient to the corresponding extension for the Glauberman correspondent across the fixed point subgroup. This theorem is essential for passing from local (fixed-point subgroup) character data back to global ($G$) character data.

A further corollary transitions from extension to counting: for $C\le S\le D\le B$ with $S\unlhd D$,

$|\Irr_D(KS\mid\theta)| = |\Irr_D(S\mid\theta^*)|,$

so the enumeration of irreducible characters above $\theta$ inflating up from $K$ is governed by the count for the Glauberman correspondent in the fixed-point subgroup.

4. Equivariant Generalization and Central $p$-Subgroups

The strategy generalizes to equivariant settings involving a central $p$-subgroup $Z\le G$ and an $A$-fixed linear character $\lambda\in\Irr(Z)$. Given $K\unlhd G$ of order prime to $p$ with $P\in\syl pG$ normalizing $K$, and writing $B=N_A(P)$, $H=N_G(P)$, $C=\Cent_K(P)$, Theorem OW-revisited guarantees

$|\Irr_{p',A}(G\mid\theta \times \lambda)| = |\Irr_{p',B}(H\mid\theta^* \times \lambda)|,$

where $\theta\in\Irr(K)^P$ and $\theta^*$ is its Glauberman correspondent.

The proof proceeds: (a) reducing the problem via induction and Gallagher’s lemma to counting characters of types $\Irr(G\mid \widehat\theta \otimes \mu)$, (b) using Schur–Zassenhaus to split complements and applying Clifford theory for identification of character sets, and (c) applying the extension–counting correspondence recursively on subgroups involving $P$ and $K$.

This result underpins the construction of equivariant bijections crucial to the McKay correspondence with automorphisms.

5. Application to Equivariant McKay Bijections

Integrating the above elements, one obtains the following theorem (Rossi–Maltempo–Vallejo (Maltempo et al., 15 Dec 2025)): let $N\unlhd G\unlhd A$ with $G/N$ $p$-solvable, $P\in \syl p G$ stabilized by $A$, and $\mu\in\Irr(N)^P$ $P$-invariant. There exists an $A$-equivariant bijection

$\Irr_{p'}(G\mid\mu) \longleftrightarrow \Irr_{p'}(N_G(P)N \mid \mu).$

This realizes an automorphism-equivariant version of the McKay conjecture for $p$-solvable groups.

The Okuyama–Wajima argument is used at the step where one reduces to a subgroup with a central $p$-power order, allowing decomposition $N_G(P)=K\times Z$ and application of Theorem OW-revisited. The counting lemmas ensure that passing to inertia subgroups or modding out $p'$-components preserves perfect bijections, preventing spurious creation or loss of irreducible characters.

6. Structural Summary and Significance

The Okuyama–Wajima argument orchestrates a reduction–extension–counting sequence across several layers of group and character theory:

Component	Content/Result	Associated Theorem
Local–global character link	Extension of $\theta$ across $K\rtimes U/C$ iff extension of $\theta^*$ across $U/C$	Okuyama–Wajima Extension
Equivariant enumeration	$A$-fixed irreducibles counted by $\theta$-good conjugacy classes	Equivariant Gallagher
Class-function isomorphism	Bases for class-functions indexed by characters or conjugacy classes	Navarro
Full correspondence	$A$-equivariant McKay bijection for $p$-solvable groups with automorphisms	Rossi–Maltempo–Vallejo

This methodology is notable for providing a self-contained, elementary approach to the equivariant McKay conjecture for $p$-solvable groups, systematically invoking character-theoretic correspondences, handle invariance under automorphisms, and function space isomorphisms, independent of deeper geometric or cohomological machinery. The core insight—reduction of character extension and enumeration problems via the Glauberman correspondence and the fusion with generalized Gallagher lemmas—has structural consequences for the theory of finite group representations and their symmetry properties (Maltempo et al., 15 Dec 2025).