Lifting Group Characters

Suppose that G is a finite group, H a subgroup, and K an invariant subset of H. Under certain conditions, we shall describe a map a which lifts every (generalized) character X of H vanishing on H K into a character XB of G. This map will be a linear isometry, in the usual metrics. It is a generalization and simplification of a similar map introduced by Feit and Thompson (see ? 4 below). Suzuki has used another special case of this map to construct normal complements to certain Hall subgroups. Our procedure is this: First (in ? 1) we associate to each class Ki contained in K, an invariant subset (S(Ki) of G. Next (in ? 2) we define f r for every class function f on H vanishing on H K. It is that class function on G which has the same value on each (S(Ki) as f has on Ki, and is zero elsewhere. Under certain simple conditions this is easily seen to be a linear isometry of class functions (see Proposition 2). The problem is to show that a maps characters into characters. This we demonstrate by proving (in Theorem 3) that yf is actually a linear combination of certain induced class functions. Finally (in ? 4), we trace the connection with the map of Feit and Thompson mentioned earlier.