# On the rank of compact p-adic Lie groups - Mathematics > Group Theory

On the rank of compact p-adic Lie groups - Mathematics > Group Theory - Download this document for free, or read online. Document in PDF available to download.

Abstract: The rank rkG of a profinite group G is the supremum of dH, where H rangesover all closed subgroups of G and dH denotes the minimal cardinality of atopological generating set for H. A compact topological group G admits thestructure of a p-adic Lie group if and only if it contains an open pro-psubgroup of finite rank.For every compact p-adic Lie group G the rank rkG is greater than or equalto dimG, where dimG denotes the dimension of G as a p-adic manifold. Inthis paper we consider the converse problem, bounding rkG in terms of dimG.Every profinite group G of finite rank admits a maximal finite normalsubgroup, its periodic radical. One of our main results is the following. Let Gbe a compact p-adic Lie group with trivial periodic radical, and suppose that pis odd. If G has trivial p-1-torsion, then rkG = dimG.

Author: ** B. Klopsch**

Source: https://arxiv.org/