Sylow 𝑝-subgroups of the general linear group over finite fields of characteristic 𝑝

Abstract

Let eij denote the matrix with the 1 of K in the (i, j) position and 0 elsewhere; we shall call any matrix of the form 1 + Ei<j aijei5 1 -triangular. The group Gn of all 1-triangular matrices in GLn(K) is a Sylow p-subgroup of GLn(K). We shall often write G for Gn if this is unambiguous. p is assumed throughout to be an odd prime. The generators 1 +ae, i+1 and the fundamental relations connecting them are studied carefully in a recent paper by Pavlov2 (for the particular case q=p) and we have therefore mentioned them briefly in the opening paragraph. When i <j the group Pij of all 1 +aeij (a CK) is isomorphic to the additive group of K. Any subgroup P of G generated by these Pij is characterised by a partition diagram IPI. These partition diagrams bear a strong resemblance to the row of hauteurs which define the sous-groupes parallelotopiques of the Sylow p-subgroups of the symmetric groups on pn symbols, studied by Kaloujnine.3 A necessary and sufficient condition is given for the partition subgroup P to be normal in G and if P'= (P, G), P*/P = centre of G/P, the duality between P' and P* is emphasised by constructing their partition diagrams. Certain automorphisms are introduced and used to prove that any characteristic subgroup of G is a normal partition subgroup. The maximal abelian normal subgroups are fully investigated and used in conjunction with the symmetry about the second diagonal to give a simple combinatorial proof that the characteristic subgroups of G are precisely those given by symmetric normal partitions. In the last section we finally identify the group of automorphisms of G.