Conjugacy Classes Of A Class Of Generalized Symmetric Groups

Let (l, m) be an ordered pair of positive integers. Let l= {1, 2, . . . , l}, m= {1, 2, . . . , m} ands = l×m= {(1,1),...,(l,1),(1,2),...,(l,2),...,(1,m),...,(l,n)}.Denote by S_m~l the symmetric group of the set s. A subset I of s is called admissible if (i, j)∈Ithen (i', j) I for any i'∈l\i. Let (?)(l, m) be the subgroup of the symmetric group S_m~l consistingof those permutations which send admissible subsets of s into admissible subsets. The groupS(l,m) is called the generalized symmetry group of size (l,m). The subgroup consisting of allthe even permutations of the (?)(l, m) is called the generalized alternate group, and is denotedby (?)(l, m). The purpose of this paper is to study the conjugacy classes of the generalizedsymmetry group (?)(l, m). The main content is as follows:(1) Determine a necessary and su?cient condition that two permutations of (?)(l, m) con-jugate ;(2) Determine the number of the conjugacy classes of (?)(l, m) ;(3) Determine the order of a conjugacy class of (?)(l, m) ;(4) Determine a necessary and su?cient condition that two permutations of (?)(l, m) con-jugate under (?)(l, m) .