S-functions, as developed by D.E. Littlewood, are reviewed and their properties developed and extended. A computer programme has been written to perform most S-function operations. In particular, S-function division is defined, which produces many simplifications, and general methods for calculating both inner and outer plethysm are derived.

The properties of matrix groups are similarly discussed using the theory of S-functions and programmes written to produce branching rules and Kronecker products. This required some new work on spin characters and on the difference characters of even dimensional rotation groups.

Generalized Racah tensors are used to study the group properties of general mixed configurations of electrons. Some properties of factorized general coupling and recoupling coefficients are also derived. These properties are used to calculate part of general two body fractional parentage coefficients.

The above methods are used to investigate the usefulness of the group R4 as an approximate symmetry for the first row atoms. It is found that the interaction with the underlying ls2 shell completely breaks this symmetry.