The McKay conjecture with group automorphisms and the Okuyama-Wajima argument
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912765267935232 |
|---|---|
| author | Maltempo, Adele Vallejo, Carolina |
| author_facet | Maltempo, Adele Vallejo, Carolina |
| contents | 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 $θ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_13406 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The McKay conjecture with group automorphisms and the Okuyama-Wajima argument Maltempo, Adele Vallejo, Carolina Representation Theory Group Theory 20C15 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 $θ$. |
| title | The McKay conjecture with group automorphisms and the Okuyama-Wajima argument |
| topic | Representation Theory Group Theory 20C15 |
| url | https://arxiv.org/abs/2512.13406 |