Generalized n-abelian groups

Abstract

Let n be an integer >= 2. A group G is called generalized n-abelian if it admits a positive polynomial endomorphism of degree n, that is if these exist n elements a(1), a(2),..., a(n) of G such that the function phi : x -> x(a1)x(a2) ... x(an) is an endomorphism of G. In this paper we give some sufficient conditions for a generalized n-abelian group to be abelian. In particular, we show that every group admitting a positive polynomial monomorphism of degree 3 is abelian.