Cycle Index of Internal Direct Product Groups

Abstract

If M and H are permutation groups with cycle indices ZM and ZH respectively, and if * is some binary operation on permutation groups, then a fundamental problem in enumerative combinatorics is the determination of a formula for ZM *H in terms of ZM and ZH. To this end, a number of results have already been obtained (cf. Harary [1], [2], [3]; Harary and Palmer [6]; Harrison and High [7]; Pόlya [10]). This paper may be viewed as a continuation of a previous paper (Kamuti [8]) in which I have shown how the cycle index of a semidirect product group G= M×H can be expressed in terms of the cycle indices of M and H by considering semidirect products called Frobenius groups. Thus if G=M×H (internal direct product), the aim of this paper is to express the cycle index of G in terms of the cycle indices of M and H when G acts on the cosets of H in G.