, Volume 96, Issue 4, pp 321-333

On the rank of compact p-adic Lie groups

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Abstract

The rank of a profinite group G is the basic invariant ${{\rm rk}(G):={\rm sup}\{d(H) \mid H \leq G\}}$ , where H ranges over all closed subgroups of G and d(H) denotes the minimal cardinality of a topological generating set for H. A compact topological group G admits the structure of a p-adic Lie group if and only if it contains an open pro-p subgroup of finite rank. For every compact p-adic Lie group G one has rk(G) ≥ dim(G), where dim(G) denotes the dimension of G as a p-adic manifold. In this paper we consider the converse problem, bounding rk(G) in terms of dim(G). Every profinite group G of finite rank admits a maximal finite normal subgroup, its periodic radical π(G). One of our main results is the following. Let G be a compact p-adic Lie group such that π(G) = 1, and suppose that p is odd. If $\{g \in G \mid g^{p-1}=1 \}$ is equal to {1}, then rk(G) = dim(G).