A note on multiplicity-free permutation characters

Abstract The transitive permutation character (1 H ) G , where G is a group and H ⩽ G , is said to be multiplicity-free if each of its irreducible constituents occurs with multiplicity one. The following result, inspired by Gelfand's (1950) work on Riemannian symmetric spaces, and also obtained by Kawanaka and Matsuyama (1990), is proved using a different method: Let G be a group of odd order and τ an involutory automorphism of G. Let H = ( g ϵ G | g τ = g ). Then (1 H ) G is multiplicity-free .