We establish a new characterization of primitive elements in free groups, which is based on the distributions they induce on finite groups.

More specifically, for every finite group G, a word w in the free group on k generators defines a word map from G^k to G. We say that w is measure preserving with respect to G if given uniform distribution on G^k, the image of this word map distributes uniformly on G. It is easy to see that primitive words (words which belong to some free generating set of the free group) are measure preserving w.r.t. all finite groups, and several authors have conjectured that the two properties are, in fact, equivalent. In a joint work with O. Parzanchevski, we prove this conjecture. The proof is based on the average number of fixed points in a random permutation distributed according to w. All notions will be defined during the talk.