The degree of a linear homogeneous group

Abstract

The problem considered in this article is that of the relation between the degree of a linear homogeneous group and the abstract properties of the group. We have the theorem of Frobenius(l) to the effect that the degree of an irreducible group is a divisor of the order of the group, and then the generalization of this by Schur(2) which states that the order of the group is divisible by the product of the degree and the order of the central. In these Transactions(') I called attention to the importance of the commutators of the group in this connection by showing that the degree of an irreducible group is not less than the order of any invariant commutator. As a matter of fact, it is a multiple of this order. It is shown in the present paper that the orders of certain non-invariant commutators also have a bearing on the degree of a'linear group, whether it be reducible or irreducible. Moreover for groups of a certain category the degree is a multiple of the order of the whole commutator subgroup. If a linear homogeneous group has a similarity substitution as a commutator, the degree of the group is a multiple of the order of this commutator since all the multipliers of a similarity substitution are equal and the determinant of a commutator is equal to one. If G is an irreducible group of order g and degree n with a central of order a all its invariant substitutions are similarity substitutions, and if ,B is the order of an invariant commutator j32 is a divisor of g since g is divisible by na and a and n are both divisible by /. If G is reducible at least one of its irreducible components has an invariant commu'tator whose order is equal to the highest order of any multiplier of any invariant commutator of G. Therefore the degree of this component is a multiple of this order, and the degree of G is greater than this order. Hence we have the following theorem.