Subgroups of Order a Power of p in the General and Special m-ary Linear Homogeneous Groups in the GF[p n

Abstract

1. It would seem that the most effective method of determining all the subgroups of order a multiple of p of a l'inear group in the Galois field of order pn is that based upon a complete knowledge of the subgroups of order a power of p. This method has proved successful for the ternary groupsl and, as I will show on another occasion, also for the quaternary groups. The present investigation proceeds far enough to give a clear insight into the nature of the simple laws pervading the subject. It is hoped that the results are capable of extension by induction to all powers of p. To indicate the difficulty of this step, it may be remarked that its completion would give the means of deriving at once an explicit list of all groups of order a power of a prime, and simultaneously all the subgroups of each. A second aim of the paper was to furnish data for the problem of the determination of all m-ary groups for low values of m. Following Lie, I write (A, B) for the commutator A-B1AB. I employ the usual notation Bj for the transformation which alters only i, replacing it by 6; + afj. We have the simple relation ?