The pro-p group of upper unitriangular matrices

Abstract

We study the pro-p group G whose finite quotients give the prototypical Sylow p-subgroup of the general linear groups over a finite field of prime characteristic p. In this article, we extend the known results on the subgroup structure of G. In particular, we give an explicit embedding of the Nottingham group as a subgroup and show that it is selfnormalising. Holubowski [13–15] studies a free product Cp⁎Cp as a (discrete) subgroup of G and we prove that its closure is selfnormalising of infinite index in the subgroup of 2-periodic elements of G. We also discuss change of rings: field extensions and a variant for the p-adic integers, this latter linking G with some well known p-adic analytic groups. Finally, we calculate the Hausdorff dimensions of some closed subgroups of G and show that the Hausdorff spectrum of G is the whole interval [0,1] which is obtained by considering partition subgroups only.