On the groups which have the same group of isomorphisms

Abstract

The main object of this paper is the determination of all the possible groups whose group of isomorphisms is either the symmetric group of order 6 or the synimetric group of order 24. We shall also determine the infinite system of groups whose group of cogredient isomorphisms is the former of these two symmetric groups. It will be proved that this system includes one and only one group (which is not the direct product of an abelian and a non-abelian group) for every power of 2. It is well known that every simple isomorphism of a group G with itself may be obtained by transforming G by means of operators that transform it into itself.t In what follows we shall generally employ this method of making G simply isomorphic with itself. In a few cases it will be convenient to employ two special methods, which 'we proceed to explain. The first of these two methods may be employed when G contains a subgroup H' which is composed entirely of operators which are selfconjugate under G and which is also simply isomorphic to a quotient group of G with respect to a selfconjugate subgroup which includes H'. In this case we may evidently multiply all of the operators of each one of the various divisions of G with respect to this quotient group by the corresponding operator of H' and thus obtain a simple isomorphisin of G with itself.-To illustrate this method we may employ the direct product G12 of' the symmetric group of order 6 anid an operator s1 of order two. If 'we multiply each of the six operators of G12 which are not contained in its cyclical subgroup of order 6 by s1 we obtain a simple isomorphism of G12 with itself. It is evident that this isomorphism corresponds to the selfconjugate operator of order two in the group of isomorphisms of G12 t It is important to observe that any operator t1 of the group of isomorphisms of G which is obtailned in this manner is selfconjugate under this group of isomorphisms whenever H' is composed of characteristic operators