THE CONJUGACY CLASSES OF A SUBGROUP $S_{n}^{m} : C_{m}$ OF Smn, prime m

Abstract

The conjugacy classes of any group are important since they reflect some aspects of the structure of the group. The construction of the conjugacy classes of finite groups has been a subject of research for several authors. Let n,m be positive integers and [Formula: see text] be the direct product of m copies of the symmetric group Sn of degree n. Then [Formula: see text] is a subgroup of the symmetric group Smn of degree m × n. Let g∈Smn, of type [mn] where each m-cycle contains one symbol from each set of symbols in that order on which the copies of Sn act. Then g permutes the elements of the copies of Sn in [Formula: see text] and generates a cyclic group Cm = 〈g〉 of order m. The wreath product of Sn with Cm is a split extension or semi-direct product of [Formula: see text] by Cm, denoted by [Formula: see text]. It is clear that [Formula: see text] is a subgroup of the symmetric group Smn. In this paper we give a method similar to coset analysis for constructing the conjugacy classes of [Formula: see text], where m is prime. Apart from the fact that this is an alternative method for constructing the conjugacy classes of the group [Formula: see text], this method is useful in the construction of Fischer–Clifford matrices of the group [Formula: see text]. These Fischer–Clifford matrices are useful in the construction of the character table of [Formula: see text].