Vertices and normal subgroups in solvable groups

Abstract

For a finite group G and a fixed prime p, one can attach to each irreducible Brauer character φ of G a p-subgroup Q, called the vertex of φ, that is unique up to conjugacy. In this paper we examine the behavior of the vertices of φ with respect to normal subgroups when G is assumed to be p-solvable. For arbitrary finite p-solvable groups, we develop a correspondence between the set of Brauer characters of G with vertex Q and the set of Brauer characters of a certain subgroup of G with vertex Q. Moreover, in the case that G has odd order, we extend a result of Navarro regarding the behavior with respect to normal subgroups of a correspondence of Brauer characters of p ′ -degree to a result regarding the behavior with respect to normal subgroups of a correspondence of Brauer characters of arbitrary degree.