Third Term of the Lower Autocentral Series of Abelian Groups

Abstract

Let G be a group and Aut(G) be the group of auto- morphisms of G. Then [g, α, β] = (g−1gα)−1(g−1gα)β is the au- tocommutator of the element g ∈ G and α, β ∈ Aut(G) of weight 3. Also, we define K2(G) [g,α,β] : g ∈ G,α,β ∈ Aut(G) > to be the third term of the lower autocentral series of subgroups of G. In this paper, it is shown that every finite abelian group is isomorphic to the third term of the autocentral series of some finite abelian group.