The torsion subgroup of p-adic analytic pro-p groups

Abstract

We provide a new proof of the recent result that the torsion elements form a subgroup in certain p-adic analytic groups. In particular, if p is an odd prime and G is a finitely generated p-adic analytic group such that γh(p–1)(G) ≤ G p[h], then the torsion elements form a subgroup. This result is best possible as there is a finitely generated p-adic analytic group in which γh(p-1)+1(G) ≤ G p[h] for all h≥ 1 and in which the torsion elements do not form a subgroup. Our proof uses the techniques of pro-p groups and involves much less technical detail then the original proof (though we must borrow one result from that proof ). As part of the proof we also find more information on the elements of finite order in the automorphism group of a uniformly powerful pro-p group.