Using group theory in reversible computing

The (2^w)! reversible transformations on w wires, i.e. reversible logic circuits with w inputs and w outputs, together with the action of cascading, form a group, isomorphic to the symmetric group S₂w. Therefore, we investigate the group S_n as well as one of its subgroups isomorphic to S_n/2 times S_n/2. We then consider the left cosets, the right cosets, and the double cosets generated by the subgroup. Each element of a coset can function as the representative of the coset. Different choices of the coset space and different choices of the coset representatives lead to four different syntheses for implementing an arbitrary reversible logic operation into hardware. Comparison leads to a best choice: a single coset space, with representatives that are generalized TOFFOLI and FREDKIN gates.