On imprimitive linear homogeneous groups

Abstract

1. The presenit paper is devoted first to the proof of a theorem fundamental in the construction of imprimnitive linear homogeneous groups t in a given niumber of variables. Then, by means of this and earlier theorems given by the author on the subject of linear groups, t JORDAN'S theorem, j to the effect that the order of a linear homogeneous group G in n variables is of the form Xf, where f is the order of an abelian self-conjugate subgroup of G, and X is less than a fixed number depending onlv upon n, is proved for imprimitive groups, a number being found that X must divide. Finally, the principal imprimitive collineation-groups in 4 variables are found and their generating substitutions given. THEOREM. Either an imprinzitive linear homogeneous group G can be written in monomial tfrm, ? or the n variables of the group can be so selected that they fall into k sets qf imiprinit'ivity of m variables each (n = kin), permuted according to a permutation-group K in k letters, which group is transitive (in the sense of transitivity of permutation-groups). The subgroup ( G') Of G, corresponding to the subgroup (K') of K which leaves one letter unchanged, is primitive (in the sense used in linear homogeneous groups) in the m variables of the set corresponding to the letter that K' leaves unchanged. In order that G may be transitive (as a linear homogeneous group, i. e., irreducible), it is plainly necessary that its sets of imprimitivity contain the same number of variables, and that the permnutation-group K, permuting these sets, is transitive (as a permutation-group). We shall prove that, if the