On the geometric mean of character degrees of a finite group

Abstract

Over the past decade, several interesting results on the so-called average character degree of a finite group and its connections with the group structure have been found. In this paper we introduce the geometric mean of character degrees and prove that the alternating group $\mathsf{A}_5$, which is the smallest non-solvable group, has minimal geometric mean of irreducible character degrees among all non-solvable groups.