ABSTRACT
This chapter examines the basic notions which will be used in quantum chemistry. The application of point groups is fairly well known in quantum chemistry but the symmetric group properties are less familiar. The left co-set associated with a subgroup is obtained by multiplying the elements of the subgroup from the left by an element of the group. If the element belongs to the subgroup then the left multiplication leaves the subgroup unchanged. By multiplying the subgroup by element of the group which does not belong to the subgroup generates a left co-set. This co-set has no element in common with the subgroup. The whole group can be decomposed into a set of co-sets which have no elements in common, each co-set has the same number of elements as the subgroup. The chapter shows that the double co-set has many important applications in calculation of the matrix elements of the Hamiltonian.
