The Splitting of Certain Solvable Groups1

Let G be a finite group. We shall designate the commutator subgroup of G by G2= [G, G]; this is the group generated by all commutators [g, h]=ghg~lh~1. Inductively G"=[Gn~~1, G] is defined to be the group generated by commutators of elements of G with elements of GB_1; and G* will designate HiT-i^"It should be recalled that G is nilpotent if G* =P, the subgroup consisting of the identity element, or equivalently, if G is the direct product of p-groups. Our object here is to show that when G* is Abelian then there is a nilpotent group X so that G = XG* where X(~\G* =E. If there are two such splittings of G into XG* and YG* then Y and X are conjugates by an element of G*. If x is in the center of X then x does not commute with any of its conjugates. As a consequence of the properties of the splitting it will follow that if G has no center and G* is Abelian, then both G and its group of automorphisms are contained in the holomorph of G*. We shall also give an example to show that the hypothesis that G* be nilpotent instead of Abelian is insufficient to insure a splitting of G in this fashion.