Groups in Which a Large Number of Operators May Correspond to Their Inverses

An abelian group may be defined by the property that, in an automorphism of the group, more than three fourths its operators may be placed in a one to one correspondence with their inverses.f It may be of interest to know the groups possessing the property that five eighths or more of the operators may be made to correspond to their inverses. The principal object of this paper, however, is to establish the following elementary theorem (I) and to illustrate the use that may be made of it in certain problems. Theorem I. A group that has two invariant subgroups with nothing in common but the identity can be set up as a multiple isomorphism between two groups of lower order. Let a group (G) of order kxk2x have the two invariant subgroups Kx and K2 of order h. and k2 respectively. If Kx and K2 have only the identity in common, every operator of Kx is commutative with every operator of K2. It may be assumed that G is not merely the direct product of Kx and K2. Let l,r2,r3, •■ -, be the operators of Kx and 1, «2, s3, • • ■, those of K2. Now