On Groups of Order p α q β
Proceedings of the London Mathematical Society
IT may be convenient to the reader to summarize the results hitherto obtained with regard to groups of order p a q^ other than those relating to particular values of p, q, a, and (3. If m is the index to which p belongs, mod. q, the first result arrived at was that, if a ^ m, the group is soluble.* In my book on the Theory of Groups (1897) I extended this result, showing that, if a < 2m, the group is soluble. In the same place I proved that, if the sub-groups of orders p a and q? are both
... d q? are both Abelian, the group is soluble; and that all groups of order p a q 2 are soluble. Of the last result another proof was given by Jordan (Liouville's Frobenius has shown that when a < 2m the group is soluble, and also that when the group contains only p m sub-groups of order q p it is soluble. In the present paper I have attacked the question of the solubility of a group of order p a q^ by a consideration of certain properties of the groupcharacteristics of such a group ; and I have succeeded in showing that all groups of order p a q p . are soluble. The first section of the paper is concerned with a property of the characteristics of certain operations in an irreducible group of linear substitutions in p m variables, where p is prime ; and it has bearings on other questions beside those with which the remainder of the paper is concerned.