Generalizing a theorem of Gagola and Lewis characterizing nilpotent groups

Abstract

Gagola and Lewis proved that a finite group G is nilpotent if and only if $\chi (1)^2$ divides |G :  $\mathrm{Ker}$ $\chi |$ for all irreducible characters $\chi$ of G. In this paper, we prove the following generalization that a finite group G is nilpotent if and only if $\chi (1)^2$ divides |G :  $\mathrm{Ker}$ $\chi |$ for all monolithic characters $\chi$ of G.