A Presentation And The Length Function 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}and s = l×m = {(i, j) | i∈l, j∈m}. Let S ml be the symmetric group of the set s. Wecall a subset I of s to be admissible if (i, j)∈I then (i′, j) I, ? i′∈l \ {i}. Let S (l, m)be the subgroup of the symmetric group S ml consisting of those permutations which sendadmissible subsets of s into admissible subsets. The group S (l, m) is called the generalizedsymmetry group of size (l, m), which is isomorphic to the wreath product of the symmetricgroup Sl by the symmetric group Sm. The main contents of this paper is as follows:(1) Give a presentation for S (l, m) via generators and relations ;(2) Determine the length function l(w) on S (l, m) ;(3) Determine the the length polynomial LG(t) of S (l, m) .