Some remarks on aperiodic elements in locally nilpotent groups

Abstract

In this paper we point out some situations in which one can recognize the local nilpotence of a non-periodic group looking at properties of its aperiodic part. Mathematics Subject Classification: 20F18, 20F19, 20F45 All through this paper, T (G) will denote the set of all periodic elements of a group G. Let G be a non-periodic group, and suppose that T (G) is a subgroup of G. Then G = T (G)∪X, where X is the subgroup generated by all aperiodic elements of G. Therefore G = X since G is non-periodic. This trivial fact is used here in several situations to recognize the local nilpotence of a non- periodic group looking at properties of its aperiodic part. Clearly, the following results make some sense only when the group has both periodic and aperiodic elements. Proposition 1. Let G be a non-periodic group, and let c be any positive in- teger. Then G is nilpotent of class at most c if and only if every aperiodic element of G is contained in Zc(G). Proof. Of course it suffices to prove that T (G) is a subgroup of G, which implies G = T (G) ∪ Zc(G) and G = Zc(G) since G is non-periodic. Suppose not. Hence there exist periodic elements a and b in G with ab aperiodic. Thus ab ∈ Zc(G). Then the subgroup H = � a, ab� = � a, bis nilpotent (of class at most c). Therefore H is finite and ab is periodic, a contradiction.