On group equations

Suppose f is a map from a non-empty finite set X to a finite group G. Define the map ζfG:G⟶N∪{0} by g↦|f−1(g)|. In this article, we show that for a suitable choice of f, the map ζfG is a character. We use our results to show that the solution function for the word equation w(t1,t2,…,tn)=g (g∈G) is a character, where w(t1,t2,…,tn) denotes the product of t1,t2,…,tn,t−11,t−12,…,t−1n in a randomly chosen order