On the holomorph of the cyclic group of order 𝑝^{𝑚}

Abstract

1. Intodxctton. The present article is a continuation of one pahlished in this journal, vol. 4 (1903) p. 151 under the more general titles '; On the holomorph of a cyclic group.' It was observed that the holomorph of the eyclic group of any order is the direct product of the holomorphs of its Sylow subgroups and hence the investigation of the holomorph of the general cyelic group may he divided into the consideration of the holomorph of the cyclic group of order pm and the study of the direct product of given groups. The present article is devoted to some of the important questions under the first topic, which have not been explicitly treated in the earlier paper. The holomorph of a cyclic group not only plays a £undamental role in many group theoretic discussions, but also presents questions equivalent to those arising in the theory of exponents to which nunlbers belong with respect to a given arbitrary modlllus m. For instanee, the numbers which have recently been called t the primitive roots of m are those which correspond to the operators of highest order in the group of isomorphisms of the cyclic group of order m and hence the determination of the number of such primitive roots is a special case of the determination of the number of operators of a given order in an abelian group. :t 2. Cyclic groteps of order Sms ?$ > 3 BTe begin with the case when p = 2 and shall assume that the cyclic group (E[) of order 2m, m > 3, is represented as a regular substitution group. The holomorph (S) of EF is composed of all ffie substitutions in these 2m letters which transform 1T into itself. Let sl represent a substitution § of order 2m-2 which generates all those slltstitutions of the group of isomorphisms (I) of ff which are commw;lltative with the operators of order 4 in ER From the mauller in which s, transforms the substitutions of IIs it is evident that sl is composed of 2 cyeles of each of the orders