Inductive McKay condition for finite simple groups of type C

TL;DR

This paper verifies the inductive McKay condition for finite simple symplectic groups of Lie type C, advancing the broader effort to prove the McKay conjecture for all finite groups.

Contribution

It introduces a new counting argument to determine character stabilizers and extends results to normalizers of Sylow d-tori in symplectic groups.

Findings

Confirmed the inductive McKay condition for type C groups

Developed a novel counting method for character stabilizers

Extended character analysis to Sylow d-tori normalizers

Abstract

We verify the inductive McKay condition for simple groups of Lie type C, namely finite projective symplectic groups. This contributes to the program of a complete proof of the McKay conjecture for all finite groups via the reduction theorem of Isaacs-Malle-Navarro and the classification of finite simple groups. In an important step we use a new counting argument to determine the stabilizers of irreducible characters of a finite symplectic group in its outer automorphism group. This is completed by analogous results on characters of normalizers of Sylow d-tori in those groups.