Retrieving information about a group from its character degrees or its class sizes

Abstract

We prove that a knowledge of the character degrees of a finite group G and of their multiplicities determines whether G has a Sylow p-subgroup as a direct factor. An analogous result based on a knowledge of the conjugacy class sizes was known. We prove variations of both results and discuss their similarities.

9 pages, to be published in Proc. Amer. Math. Soc