Characters and solutions to equations in finite groups

The number of ways an element of a finite group can be expressed as a square, a commutator, or more generally in the form w(x ₁, .., x _r), where w is a word in the free group, defines a natural class function. We investigate some properties of these class functions, in particular their tendency to be characters or virtual characters of the underlying group. Generalizing classical results of Frobenius and others, we prove that generalized commutators yield characters in this manner, and use this to exhibit a criterion for nilpotency based on a certain equation associated with the irreducible characters.