A characterization of generalized permutation characters

Abstract

Permutation characters and permutation modules play an important role in finite group representation theory. It is therefore desirable to know which generalized characters of a group are generalized permutation characters-that is, integral linear combinations of permutation characters. All characters of symmetric groups are generalized permutation characters, as are all rationally represented characters of p-groups. Several papers on generalized permutation characters have appeared but as far as we know ours is the first characterization of generalized permutation characters for arbitrary finite groups. See the introduction of [6] for a discussion of some recent wprk on generalized permutation characters. Our approach to the problem involves systematic use of p-integral linear combinations of irreducible characters of a group G, rather than merely integral linear combinations. Clearly a generalized character of G is a generalized permutation character if and only if it is a p-integral combination of permutation characters for all p dividing 1 G I. Before we state our main theorem we must give two definitions. For the first, let G be a group, p a prime number, and (x) a cyclic #-subgroup of G. Let x be a rational-valued class function on G, and let P, be ap-Sylow of Cc(x). We define a class function xZ on P, by x=(p) = x(px) for p E P, . For the second definition, let H be a not necessarily normal subgroup of a group G and let 9 be a class function on H. We say that 0 is G-invariant if 0 has the same value on any two G-conjugate elements of H. In the statement of the main theorem which follows, pZ will denote a p-Sylow of N&x) containing P, .