Irreducible homogeneous linear groups in an arbitrary domain

I have given elsewhere a necessary and sufficient condition that certain categories of abstract groups of finite order be simply isomorphie with irreducible homogeneous linear groups in the domain of all real and complex numbers.t In the present paper I establish a two-fold generalization of this result by showing that the same condition applies to all groups of finite order and to an arbitrary domain. The proof for the necessity of this condition rests upon a conclusion drawn from Theorem II of SCHUR'S lMeue.Begrundung der Cheorie der Gr?ppencharaktere.t. Suppose that we have a hologeneous linear group S of finite order whose coefficients belong to n (an arbitrary finite field or an arbitrary domain) and which is irreducible in . If P is a given substitution on the same / variables