Casimir-Operator und ein Satz von Reynolds über Charaktergrade

Abstract

The following result concerning character degrees is familiar. Let H be a normal subgroup of the finite group G with the identity e. Furthermore let $\theta $ be an irreducible complex character of H, and $\chi $ be an irreducible complex character of G with $\big (\theta ^G,\chi \big )_G \ne 0$ . Then the co-degree $|H|/\theta (e)$ divides the co-degree $|G|/\chi (e)$ . The usual proof rests on Schur's theory of projective representations and on Clifford's theory. The purpose of this paper is to give a more elementary proof based on the so-called Casimir operator.