Counting lifts of Brauer characters

TL;DR

This paper investigates the number of lifts of Brauer characters in p-solvable groups with odd prime p, establishing an upper bound related to the abelian vertex subgroup.

Contribution

It provides new bounds on the number of lifts of Brauer characters in p-solvable groups, extending previous results to groups with abelian vertex subgroups.

Findings

Number of lifts is at most the order of the abelian vertex subgroup Q.

Develops new results on lifts of Brauer characters in p-solvable groups.

Extends known results from groups of odd order to broader classes.

Abstract

In this paper we examine the behavior of lifts of Brauer characters in p-solvable groups where p is an odd prime. In the main result, we show that if \phi \in IBrp(G) is a Brauer character of a solvable group such that \phi has an abelian vertex subgroup Q, then the number of lifts of \phi in Irr(G) is at most |Q|. In order to accomplish this, we develop several results about lifts of Brauer characters in p-solvable groups that were previously only known to be true in the case of groups of odd order.