Non-Abelian Groups Whose Groups of Isomorphisms are Abelian

Introduction. In the appendix of Hilton's Finite Groups (1908), page 233, the question whether non-abelian group can have an abelian group of isomorphisms occurs among a few interesting questions still awaiting solution. A non-abelian group of order 64 whose group of isomorphisms is abelian and of order 128 was later constructed by G. A. Miller.t No other discussian of the problem appears in the literature of Mathematics. In fact, an exhaustive analysis of the properties of non-abelian group whose group of isomorphisms is abelian seems to involve some difficulty, for the reason that little is known concerning isomorphisms of non-abelian group. The nature of an abelian group whose group of isomorphisms is abelian was determined by G. A. Miller,+ who proved that necessary and sufficient condition that an operation of the group of isomorphisms of an abelian group be invariant under this group is that it should transform every operation of this abelian group into the same power of itself. From this it is obvious that the only abelian groups whose groups of isomorphisms are abelian are the cyclic groups. In what follows we shall develop few necessary conditions which must be satisfied by non-abelian group whose group of isomorphisms is abelian. 1. The fundamental theorem. Wle assume non-abelian group G which is restricted solely by the hypothesis that its group of isomorphisms I is abelian. Since the central quotient group of G is simply isomorphic with the group of inner isomorphisms of G, it is clear that every commutator of G is invariant under G. The group G is accordingly the direct product of its Sylow subgroups.? Since every Sylow subgroup of G must correspond to itself in any isomorphism of G with itself, we may confine ourselves to the case where G is of order pm. For convenience we shall introduce the following notation: the symbol G shall consistently denote non-abelian group whose group of isomorphisms I