Some finite solvable groups with no outer automorphisms

Abstract

A group G is complete if the center Z(G) of G is trivial and if every automorphism of G is inner. In [3], all complete metabelian finite groups were determined. They are either of order 2 or direct products of holomorphs of cyclic groups of different odd prime power orders. Here we will determine all finite groups G which have a normal abelian subgroup A with G/A nilpotent and which have no outer automorphisms. All groups considered here will be finite. We write G c ~ J f if G has an abelian normal subgroup A with G/A nilpotent. Our notation will be quite standard; see, for example, [5] or [7]. The main result is as follows.