Strongly k-Abelian Repetitions

We consider with a new point of view the notion of nth powers in connection with the k-abelian equivalence of words. For a fixed natural number k, words u and v are k-abelian equivalent if every factor of length at most k occurs in u as many times as in v. The usual abelian equivalence coincides with 1-abelian equivalence. Usually k-abelian squares are defined as words w for which there exist non-empty k-abelian equivalent words u and v such that w = uv. The new way to consider k-abelian nth powers is to say that a word is strongly k-abelian nth power if it is k-abelian equivalent to an nth power. We prove that strongly k-abelian nth powers are not avoidable on any alphabet for any numbers k and n. In the abelian case this is easy, but for k > 1 the proof is not trivial.