Subnormalizers and character correspondences in $p$-solvable groups

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.