an:04156773
Zbl 0705.20035
Asada, Mamoru
An analogue of the Levi decomposition of the automorphism groups of certain nilpotent pro-\(\ell\) groups
EN
J. Algebra 132, No. 1, 160-169 (1990).
00155628
1990
j
20F28 20E18 20F14 22E20
finitely generated nilpotent pro-\(\ell \) group; descending central series; bi-continuous automorphisms; generating set; linear \(\ell \)-adic Lie group; Levi subgroup
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.
U.Kaljulaid