Major indices, Mahonian identities and ordered generating systems (extended abstract)

A classical result of MacMahon shows that the length function and the major index are equidistributed over the symmetric group. A long standing open problem is to extend the notion of major index and MacMahon identity to other groups. A partial solution was given in [3] and [5], where this result was extended to classical Weyl groups. In this paper, it is proved that various permutation groups may be decomposed into a set-wise direct product of cyclic subgroups. This property is then used to extend the notion of major index and MacMahon identity to these groups.