Powers of characters of finite groups

LetG be a finite group and θ a complex character ofG. Define Irr(θ) to be the set of all irreducible constituents of θ andIrr(G) to be the set of all irreducible characters ofG. Thecharacter-covering number of a finite groupG, 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 ofG is infinite. In this article we show that a finite nontrivial groupG has a finite character-covering-number if and only ifG is simple and non-abelian and ifG is a nonabelian simple group thenccn(G) ≤ k² - 3k + 4, wherek is the number of conjugacy classes ofG. 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,J₁. These results are analogous to results obtained previously concerning the covering of groups by powers of conjugacy classes. Other related results are shown.