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

The Probability Distribution of Word Maps on Finite Groups