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- W2076046670 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)." @default.
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- W2076046670 date "2011-04-01" @default.
- W2076046670 modified "2023-10-07" @default.
- W2076046670 title "On the rank of compact p-adic Lie groups" @default.
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- W2076046670 doi "https://doi.org/10.1007/s00013-011-0240-7" @default.
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