Analytic pro-\(p\) groups and their Lie algebras
Let \(G\) be a uniform pro-\(p\)-group. Then \(G\) can be also considered as an intrinsic Lie algebra (over \(\mathbb{Z}_p\)) with the Lie operations being defined as follows: \(g+h=\lim (g^{p^n} h^{p^n})^{p^{-n}}\), \([g,h]=\lim (g^{p^n}, h^{p^n})^{p^{-2n}}\). Conditions on a uniform group \(G\) such that for every subgroup \(H\) of \(G\), \(H\) is a subalgebra of \(G\) and for every subalgebra \(H\) of \(G\), \(H\) is a subgroup of \(G\) are found in the paper. The most interesting results are Theorem 1. Let \(G\) be an infinite uniform pro-\(p\)-group, viewed also as a Lie algebra over \(\mathbb{Z}_p\) with the operations as above, and let \(H\) be a closed subgroup of \(G\). (\(H \leq G^2\) if \(p=2\).) Then the following are equivalent: (1) Every closed subgroup \(K\) of \(G\) is also a closed subalgebra of \(G\) with the Lie algebra structure. (2) \(H\) is power closed, i.e. every closed subgroup \(K\) of \(H\) satisfies \(K^{\{p\}}=K^p\), where \(K^{\{p\}}\) means the subset of \(p\)-th powers of \(K\) and \(K^p\) means the closed subgroup generated by all \(p\)-th powers of \(K\) (if \(p=2\) this condition is replaced by: Every closed subgroup \(K\) of \(H\) satisfying \(K^{\{4\}}\) is a uniform subgroup of \(H\)). If \(H\) is power closed, then in addition, every closed normal subgroup of \(H\) is an ideal of \(H\) with the Lie algebra structure. Theorem 2. Let \(G\) be a uniform group (which has also a Lie algebra structure over \(\mathbb{Z}_p\)) such that for every \(x, y \in G\), \(p \geq \dim (\overline {\langle x, y\rangle})\). Then the set of closed subgroups of \(G\) coincides with the set of closed subalgebras of \(G\), and the set of closed normal subgroups of \(G\) coincides with the set of closed ideals. In particular, the theorem holds if \(p \geq \dim (G)\). Theorem 3. For every prime \(p\) there is a uniform group \(G\) of dimension \(p+1\), and there exist \(x, y \in G\) such that the closed Lie algebra generated by \(x\) and \(y\) is not a subgroup of \(G\), and the closed subgroup generated by \(x\) and \(y\) is not a subalgebra of the Lie algebra \(G\).
scientific article; zbMATH DE number 802368
scientific article