Sylow 𝑝-subgroups of the classical groups over finite fields with characteristic prime to 𝑝

Abstract

If S,, is a Sylow p-subgroup of the symmetric group of degree pn, then any group of order pn may be imbedded in Sn. We may express Sn as the complete product' C o C o ... o C of n cyclic groups of order p and the purpose of this paper is to show that any Sylow psubgroup of a classical group (see ?1) over the finite field GF(q) with q elements, where (q, p) = 1, is expressible as a direct product of basic subgroups En-C O C O ... o C (n factors), where Z is cyclic of order pr. (We assume always that p ;2.) Since C may be imbedded in S., we see that n is imbedded in Sn+r-l in a particularly simple way. The above r is defined by the equation q -1 =pt *where qI is the first power of q which is congruent to 1 mod p and * denotes some unspecified number prime to p. The case r = 1 is therefore of frequent occurrence, and then clearly SnSn Professor Philip Hall was my research supervisor in Cambridge (England) during the years 1949-1952 and it is a pleasure to acknowledge here my indebtedness to him for his generous encouragement.