Closed subgroup of profinite group
WebDec 3, 2009 · Profinite groups. Definition A profinite group is a compact Hausdorff topological group whose open subgroups form a base for the neighbourhoods of the … WebApr 17, 2024 · The finite groups are considered as compact discretetopological groupsand so the inverse limit, as a closed subspace of the compact space that is the product of all those finite groups has the inverse limit topology, hence is, as is said above, a compact Hausdorff, totally disconnected group. Examples Example A finite groupis profinite.
Closed subgroup of profinite group
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WebMar 10, 2010 · Profinite groups are Galois groups. As such they are of interest in algebraic number theory. Much of recent research on abstract infinite groups is related to profinite groups because... WebDec 23, 2016 · Theorem 1 asserts that in a finitely generated prosoluble group, every subgroup of finite index is open.This generalises an old result of Serre about pro-p …
WebJan 12, 2024 · Because the collection of subgroups S u b ( G i) of a finite group G i is finite, and the closed subgroups of a profinite group G = l i m G i can be described as inverse limits of subgroups of the G i, we have that S u b ( … WebThe point is that, among locally compact groups, there are two extremes: Lie groups, which have no small subgroups, i.e. sufficiently small neighbourhoods of 1 contain no non-trivial subgroup; and profinite groups, in which every neighbourhood of the identity contains an open subgroup.
WebFeb 19, 2024 · Abstract. A locally compact group G is called compactly ruled if it is a directed union of compact open subgroups. We denote by {\mathcal {SUB}}\left ( G\right) the space of closed subgroups of G equipped with the Chabauty topology. In this paper, we show that the subspace {\mathcal {SUB}}_ {\scriptscriptstyle \mathrm {co}}\left ( … WebJun 23, 2024 · Let $G$ be a profinite group and $H \unlhd G$ a closed normal subgroup. If $H$ is finite then there exists $U \unlhd G$ open normal subgroup such that $U \cap H …
WebJul 1, 2000 · CLOSED SUBGROUPS OF PROFINITE GROUPS Published online by Cambridge University Press: 01 July 2000 DAN SEGAL Article Metrics Get access Share …
WebApr 10, 2024 · Probabilistic properties of profinite groups. Let be a class of finite groups which is closed for subgroups, quotients and direct products. Given a profinite group … play from headphonesWebApr 11, 2024 · is the class of finite soluble groups or of finite nilpotent groups. In particular, for the above classes, the profinite $\mathfrak C$-positive groups are virtually prosoluble (re primary upd8WebFeb 1, 2015 · The group of $p$ -adic integers $\mathbf {Z}_p$ is a profinite group in which all subgroups of finite index are closed and open. You can show this by finding explicitly all these subgroups, but how could you show it "conceptually"? Share Cite edited Jul 2, 2024 at 9:14 answered Feb 1, 2015 at 15:00 Olórin 11.7k 1 23 47 primary university south africaWebDec 14, 2016 · Profinite group is a closed subgroup of product of finite groups (discrete topology) with product topology. We want to see Gal(F) as subgroup of product of finite groups. Consider finite galois extensions K / F, we have F … primary upcWebindex of a closed subgroup of a profinite group. In the book "profinite groups, arithmetic, and geometry" of Shatz, the index ( G: H) of a closed subgroup H of a … play from hereWebApr 10, 2024 · Probabilistic properties of profinite groups. Let be a class of finite groups which is closed for subgroups, quotients and direct products. Given a profinite group and an element , we denote by the probability that and a randomly chosen element of generate a pro- subgroup. We say that a profinite group is -positive if for all %Moreover we say ... primary university wayneWebApr 10, 2024 · Download Citation Probabilistic properties of profinite groups Let $\\mathfrak C$ be a class of finite groups which is closed for subgroups, quotients and direct products. Given a profinite ... primary unspecified cancer icd 10