The automorphisms of the holomorph of a finite abelian group

Abstract

group of odd order has only inner automorphisms. The group of automorphisms of the holomorph of an arbitrary group was studied by Gol'fand [3] who found some cases in which the outer automorphism group has order one or two. In the present paper I determine explicitly the outer automorphism group 0 of the holomorph H of an arbitrary finite abelian group G. If G is the direct product of a group of odd order, a group of order two, and a cyclic group of order 2n where n> 2, then 0 is the direct product of a finite number of groups of order two and a non-abelian group 0* of order six or eight. If n =2 then 0 * is isomorphic to the symmetric group of order six, and if n >3 then 0* is the octic group. In all other cases 0 is either trivial or the direct product of a finite number of groups of order two. Let A be the group of all automorphisms of the finite abelian group G, let GB be the group of all automorphisms of H that map G onto itself, and let g be the group of all inner automorphisms of H. Then (B3/J can be identified with the first cohomology group H'(A, G). Thus H'(A, G) can be regarded as a subgroup of 0. Now G is an ilnvariant subgroup of H, and it is known [6] that H has at most four invariant subgroups isomorphic to G. It follows that H'(A, G) has index at most four in 0.