Subnormalizers and character correspondences in $p$-solvable groups

TL;DR

This paper proves the strong subnormalizer conjecture for certain p-solvable groups, advancing the understanding of local-global conjectures in finite group representation theory.

Contribution

It establishes the conjecture for p-solvable groups with odd p under specific conditions and introduces new properties related to the Glauberman correspondence.

Findings

Proved the strong subnormalizer conjecture for p-solvable groups when p is odd and the subnormalizer subset is a subgroup.

Confirmed the conjecture for groups with p-length 1 when p is odd.

Derived new properties related to the Glauberman correspondence.

Abstract

A new family of local-global conjectures in the representation theory of finite groups has recently been proposed by Moret\'o. We show that one of the strongest of these conjectures, the strong subnormalizer conjecture, holds for $p$-solvable groups when $p$ is odd, under the condition that the subnormalizer subset is a subgroup. We also prove it in general when $p$ is odd and the $p$-length of the group is 1 and, in the process, obtain new properties related to the Glauberman correspondence.