Arch. Math. 90 (2008), 490–492 c  2008 Birkh¨auser Verlag Basel/Switzerland 0003/889X/060490-3, published online 2008-05-15 DOI 10.1007/s00013-008-2605-0 Archiv der Mathematik Abelian groups as autocommutator groups Codrut ¸a Chis ¸, Mihai Chis ¸, and Gheorghe Silberberg Abstract. In this article, we establish that every finite abelian group is isomorphic to the autocommutator subgroup of some finite abelian group. Mathematics Subject Classification (2000). 20D45, 20D25. Keywords. Autocommutator subgroup, finite abelian group. 1. Preliminaries. For a group G, an element g ∈ G and an automorphism α ∈ Aut(G), one denotes by [g,α] := g -1 g α the commutator of the element g and the automorphism α. Also, let K(G) := 〈[g,α]| g ∈ G, α ∈ Aut(G)〉 denote the autocommutator subgroup of G. Recently, P. Hegarty ([2] and [3]), proved that for any given finite group K, the number n(K) of the finite groups G such that K(G) ∼ = K is finite. M. Deaconescu and G. Walls determined in [1] all groups G having a cyclic autocommutator subgroup either infinite or of prime order. Since there are finite groups K, as for instance K = S 3 , such that n(K) = 0, a natural question is to determine those groups K satisfying n(K) ≥ 1. The aim of this note is to prove the following theorem: Theorem 1.1. Every finite abelian group is the autocommutator subgroup of some finite abelian group. All groups in the sequel will be finite; the unexplicited notation is standard and follows that of [5]. 2. Proof of Theorem 1.1. The proof is divided into several steps. 1. Let G be a group and H a characteristic subgroup of G such that [G : H]=2. Then K(G)  H. Let α ∈ Aut(G) and g ∈ G be arbitrary. If g ∈ H, then g α ∈ H and [g,α] ∈ H. If g ∈ H, then g α ∈ H, so that gH = G \ H = g α H and again [g,α]= g -1 g α ∈ H.