The Probability Distribution of Word Maps on Finite Groups

Word maps provide a wealth of information about finite groups. We examine the connection between the probability distribution induced by a word map and the underlying structure of a finite group. We show that a finite group is nilpotent if and only if every surjective word map has fibers of uniform size. Moreover, we show that probability distributions themselves are sufficient to identify nilpotent groups, and these same distributions can be used to determine abelian groups up to isomorphism. In addition we answer a question of Amit and Vishne.