Localization of q-abelian groups (Q1111698)

A group G is called n-abelian if \((xy)^ n=x^ ny^ n\) for all \(x,y\in G\). As first exploited by \textit{R. Baer} [in Proc. Am. Math. Soc. 4, 15-26 (1953; Zbl 0050.022)] many concepts and constructions can be generalized from abelian groups to n-abelian groups. The paper shows that, if q is a prime and G is a q-abelian group, then a universal homomorphism from G to a group in which qth roots exist and are unique can be obtained by taking the direct limit in the system \(G\to^{q}G\to^{q}G\to^{q}...\), exactly as in the abelian case. The key point in the proof is an argument which decides in general whether a given P-localization functor in a variety of groups agrees with the restriction of the P-localization functor in the whole category of all groups. As preparatory material, the paper contains several new results on the structure of n-abelian groups (together with a proof of the fact that the elements of finite order form a subgroup) and a practical procedure to obtain nontrivial examples of noncommutative n-abelian groups.