Third term of the lower 3-autocenteral series of abelian groups

Let G be a group and Aut(G) be the group of automorphisms of G. Th en [g,α^3,β^3] = (g^(-1)α^3(g))^(-1)β^3(g^(-1)α^3(g)) is the autocommutator of the element g ∈ G and α,β ∈ Aut(G) of weight 3. Also, the auther defines K^3_2 (G) = {[g,α^3,β^3] : g ∈ G,α,β ∈ Aut(G)} to be the third term of the lower 3-autocenteral series of subgroup of G. In this paper the auther investigates the properties of third term of the 3-autocenteral series of a group G.