On the group of isomorphisms of a certain extension of an abelian group

Abstract

In 1908 Professor G. A. Miller showed that if an abelian group H which involves operators whose orders exceed 2 is extended by means of an operator of order 2 which transforms each operator of H into its inverse, then the group of isomorphisms of this extended group is the holomorph of H.t The present paper discusses an elaboration of the idea embodied in Professor Miller's thoerem, the successive developments taking an abelian group H each of more general type so that in toto it is proved that if G is formed by extending H which has operators of order > 2 by a certain operator from its group of isomorphisms which transforms every one of its operators into the same power of itself and which is commutative with no operator of odd order in it, then the group of isomorphisms of G is the holomorph of H, and is a complete group if H is of odd order. If H contains no operators of even order, the certain operator is any operator from the group of isomorphisms of H effecting the stated automorphism; if H contains no operator of odd order, the certain operator must transform every operator of H into its inverse; if H contains operators of both even and odd orders (a) the order of the automorphism of the operators of odd order effected by the extending operator is to be divisible by 2n where H contains an operator of order 2n but none of order 2n+1, or else (b) the extending operator transforms into its inverse every one of H's operators whose order is a power of 2. Obviously it is not necessary that the extending operator be from its group of isomorphisms, but that it have certain properties possessed by this operator; thus, besides effecting the automorphism required, its first power commutative with all the operators of H must be the identity (from which follows that its first power appearing in H is the identity). The general method used in establishing each of the successive theorems is the same; two important steps in each proof are showing that H is character-