Powers of characters of finite groups

Abstract

Let G be a finite group and θ a complex character of G . Define Irr(θ) to be the set of all irreducible constituents of θ andIrr( G ) to be the set of all irreducible characters of G . The character-covering number of a finite group G , ccn( G ), is defined as the smallest positive integer m such thatIrr(χ m ) =Irr( G ) for allχ∈Irr( G )—{1 G }. If no such positive integer exists we say that the character-covering-number of G is infinite. In this article we show that a finite nontrivial group G has a finite character-covering-number if and only if G is simple and non-abelian and if G is a nonabelian simple group thenccn( G ) ⩽ k 2 − 3 k + 4, where k is the number of conjugacy classes of G . Then we show (using the classification of the finite simple groups) that the only finite group with a character-covering-number equal to two is the smallest Janko's group, J 1 . These results are analogous to results obtained previously concerning the covering of groups by powers of conjugacy classes. Other related results are shown.