Two results on the character degree sums

Abstract

Let G be a finite group, and T(G) be the sum of all complex irreducible character degrees of G. In this paper, we aim to characterize the structure of finite groups in terms of T(G). We show that if |G|/T(G)<(p+1)/2, then G has a normal Sylow p-subgroup; and that if |G|/T(G)<3p2/(p2+2), then G is p-supersolvable, where p is a prime.