A Note on Lifting Brauer Characters

A Brauer character of a finite group may be lifted to an ordinary character if it lies in a block whose defect groups are contained in a normal p-solvable subgroup. By the Fong-Swan theorem [2, Theorem 72.1], an irreducible Brauer character of a finite p-solvable group G may be lifted to an ordinary (complex) character of G. In other words, every Brauer character is the restriction of some ordinary character y to the p-regular elements of G. Professor I. M. Isaacs has shown [5] that the character y may be chosen to satisfy certain extra conditions which when p is odd, uniquely determine y. By extending a theorem which appears in that paper 15, Theorem 3.1], the hypothesis of p-solvability on G may be weakened somewhat. Specifically, the main result of this paper is the following Theorem. Let lies in a block whose defect groups are contained in a normal p-solvable subgroup of G. Then and fi, respectively, and if V is irreducible, then the multiplicity of fi in is the multiplicity of V as a composition factor of U. Lemma 1. Let N <] G and let W be an irreducible F\N\modulc which affords the Brauer character fi. Assume that fi can be lifted to an ordinary character f in such a way that the inertia groups i)r(/z) and S^CP) coincide. Denote by [L the Brauer character which the induced module W affords. Let o denote the set of all irreducible Brauer characters (f> of G which are constituents of \l , but which are not afforded by any composition factor of j{W ). Finally, let J denote the set of ordinary irreducible characters y of G which are constituents of *{* ' and which have the property that the Received by the editors December 24, 1974. AMS (MOS) subject classifications (1970). Primary 20C20.