On the characters of 𝑝-solvable groups

Abstract

Introduction.In the theory of group characters, modular representation theory has explained some of the regularities in the behavior of the irreducible characters of a finite group; not unexpectedly, the theory in turn poses new problems of its own.These problems, which are asked by Brauer in [4], seem to lie deep.In this paper we look at the situation for solvable groups.We can answer some of the questions in [4] for these groups, and in doing so, obtain new properties for their characters.Finite solvable groups have recently been the object of much investigation by group theorists, especially with the end of relating the structure of such groups to their Sylow /»-subgroups.Our work does not lie quite in this direction, although we have one result tying up arithmetic properties of the characters to the structure of certain /»-subgroups.Since the prime number p is always fixed, we can actually work in the more general class of /»-solvable groups, and shall do so.Let © be a finite group of order g = pago, where p is a fixed prime number, a is an integer ^0, and ip, go) = 1.In the modular theory, the main results of which are in [2; 3; 5; 9], the characters of the irreducible complex-valued representations of ©, or as we shall say, the irreducible characters of ®, are partitioned into disjoint sets, these sets being the so-called blocks of © for the prime p.Each block B has attached to it a /»-subgroup 35 of © determined up to conjugates in @, the defect group of the block B. If 35 has order pd, in which case we say B has defect d, and if Xn is an irreducible character in B, then the degree of Xk-¡s divisible by p to the exponent a -d+e", where the nonnegative integer e" is defined as the height of Xm-Now let © be a /»-solvable group, that is, © has a composition series such that each factor is either a /»-group or a /»'-group, a /»'-group being one of order prime to p.The following is then true: Let B be a block of © with defect group 3).If 35 is abelian, then every character Xu in B has height 0. Conversely, if B is the block containing the 1-character, and if every character in B has height 0, then 35 is abelian.In particular, this gives a necessary and sufficient condition for the Sylow /»-subgroups of © to be abelian.In the general modular theory of finite groups, each irreducible character X" can be decomposed into a sum of irreducible modular characters <£p of ®, Xn = 53 ¿nor-