Irreducible linear homogeneous groups whose orders are powers of a prime

Abstract

The purpose of this article is to determine as far as possible the connection between the degree of an irreducible linear homogeneous group and its abstract group properties. The discussion will be limited for the mnost part to groups whose orders are powers of a prime. As a particular phase of the general subject there arises the question as to the circumstances under which a group can be simply isomorphic with irreducible groups of different degrees. The simple group of order 168, for example, is simply isomorphic with irreducible groups of degrees 3, 6, 7, and 8 respectively. On the other hand, I know of no group whose order is a power of a prime that is simply isomorphic with irreducible groups of different degrees. In fact, the following discussion shows that in certain irreducible groups the degree is uniquiely fixed by certain abstract properties of the group; and if the degree is not thus uniquely fixed the order of the group must be greater than the seventh power of a prime. THEOREM I. A linear honmogeneous group G, of classt k, all of whose invariant operations are similarity substitutions either is irreducible or is simply isomorphic with each of its irreducible components. lf G is reducible, suppose that it has been put into its completely reduced formii. The invariant operations will niot be affected by this change. If in alny operation the identity of any irreducible component were associated with a nonidentical operation in the other variables, this operation would be non-invariant in G., and would therefore give at least one non-identical commutator. This comnmutator would also be non-invariant, and none of its successive commutators, except identity, could be invariant. But this is impossible since G is of class k. Hence every irreducible component of G is simply isomorphic with G. THEOREM II. A linear honTogeneous group G, qf order a power of a prime, that has a cyclic central either is irreducible or is simply isomorphic with at leaist one of its irreducible components.