Counting lifts of Brauer characters

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 ϕ\in IBrp(G) is a Brauer character of a solvable group such that ϕhas an abelian vertex subgroup Q, then the number of lifts of ϕin 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.

A different proof of Theorem 1 is in the paper "The number of lifts of Brauer characters with a normal vertex" by J.P. Cossey, M.L.Lewis, and G. Navarro. Hence, we do not expect to try to publish this note. We feel that the proof in this paper is of independent interest