The Real Representation of the Commutator S−1 T−1 ST in Four Dimensions

Abstract

It has been remarked in a former paper that perhaps a fuller understanding of the theory of linear groups may be arrived at by considering the real representations. It is sufficient that these be irreducible in the real field. In the present paper we continue this investigation and deal with the angles of rotation; in particular we find the form of the commutatorwhere S and T are substitutions of order p1 and p2 and have angles of rotation θ, θ′, θ″, … and φ, φ′, φ″, …. If certain conclusions regarding the orders of S, T, and C may be drawn we shall then be able to attack a very important problem—;that of Primitivity. This is in essence very similar to the method of Blichfeldt, since the commutator is the product ofwhich are both of order p2.