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An analogue of the Levi decomposition of the automorphism groups of certain nilpotent pro-$$\ell$$ groups. (English) Zbl 0705.20035
Let G be a finitely generated nilpotent pro-$$\ell$$ group, $$\{G_ k|$$ $$k\in {\mathbb{N}}\}$$ its descending central series with $$G_ k/G_{k+1}$$ being a free $${\mathbb{Z}}_{\ell}$$-module of finite rank k. Denote by m the least integer with $$G_ m=(1)$$. The author proves Theorem: for $$\ell \geq m$$ and the group $$\Omega$$ of bi-continuous automorphisms of G the short exact sequence $$1\to Ker \sigma \to \Omega^{\sigma}\to Aut(G/G_ 2)\to 1$$ (with $$\sigma$$ being the canonical homomorphism) splits. He notices also that there exists an automorphism $$\sigma_{\alpha}\in \Omega$$ such that $$x_ i^{\sigma_{\alpha}}=x_ i^{\alpha}$$ (1$$\leq i\leq r)$$ for a given generating set $$\{x_ 1,...,x_ r\}$$ of G and $$\alpha \in {\mathbb{Z}}^*_{\ell}$$ satisfying $$\alpha^ j\neq 1$$ (1$$\leq j\leq m-2)$$. The author shows that the centralizer $$C_{\Omega}(\sigma_{\sigma})$$ is independent of $$\alpha$$ and this subgroup $$\Pi =C_{\Omega}(\sigma_{\alpha})$$ is such that $$\Pi\cap Ker \sigma =(1)$$ and $$Im(\sigma |_{\Pi})=Aut(G/G_ 2)$$. The author notices also that $$\Omega$$ can be viewed as a linear $$\ell$$-adic Lie group and $$C(\sigma_{\alpha})$$ as its Levi subgroup. Two remarks are added: (1) for $$m>\ell$$ the above theorem isn’t true in general, and (2) there exist hopes to give some application of the theorem to Galois representations.
Reviewer: U.Kaljulaid

##### MSC:
 20F28 Automorphism groups of groups 20E18 Limits, profinite groups 20F14 Derived series, central series, and generalizations for groups 22E20 General properties and structure of other Lie groups
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