Closed subgroup of profinite group
WebEach profinite group G has a p-Sylow group Gp. By definition, Gp is a closed subgroup of G which is pro- p and which is maximal with this property. Every pro- p subgroup of G is contained in a p -Sylow subgroup and every two p -Sylow groups of G are conjugate. WebA residually finite (profinite) group is just infinite if every non-trivial (closed) normal subgroup of is of finite index. This paper considers the problem of determining whether a (closed) subgroup of a just infin…
Closed subgroup of profinite group
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WebJun 24, 2024 · A p - Sylow subgroup (or p -Sylow for short) of a profinite group G is a closed subgroup H of G which is a pro- p -group and such that the index [ G : H] is prime to p. The proposition below follows from the properties of indexes and of Sylow subgroups. Its proof uses the following easy lemma ( [6], Sect. 9.6, Prop. 8). Lemma 4.9 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 7, 2024 · A profinite group G has rank r = rk (G) if every closed (equivalently, open) subgroup can be (topologically) generated by r elements, and r is the least such integer … WebMoreover, if G is a profinite group and H ⊂G is a closed subgroup, then H is profinite. Similarly, if N ⊂G is a closed normal subgroup, then G/N is profinite with the quotient topology. It is a theorem that given a homomorphism of profinite groups f : G 1 →G 2 (in …
WebFeb 2, 2016 · 3 In general is not true that for a profinite group $G$ the derived subgroup $G'$ is closed; But if $G$ is a pro-$p$-group, with $p$ a prime, and if $G$ is finitely generated then it becomes true. For example this is true for the group of $p$-adic integer $\mathbb {Z}_ {p}$. Share Cite Follow answered Feb 19, 2016 at 23:57 The Number … WebOct 30, 2015 · Cohomology of closed subgroup of profinite group. Ask Question Asked 7 years, 4 months ago. Modified 7 years, 4 months ago. Viewed 92 times 1 $\begingroup$ I have been reading Cassels-Frohlich, and I have a question about a fact that is cited in a proof about the cohomology of pro-finite groups (the exact palace is the proof of Prop 4 …
WebA profinite group may have subgroups of finite index that are not open. For example, let G = 1 + t F p [ [ t]] (under multiplication). Then G is profinite with the subgroups 1 + t n F p [ [ t]] forming a basis of open subgroups; in particular, it …
WebDec 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 … derby college apply nowWebApr 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 … derby college apprenticeship coursesWebThis article is concerned with the representation growth of profinite groups over finite fields. We investigate the structure of groups with uniformly bounded exponential representation growth (UBERG). Using crown-base… fiberglass boat hull repair kitWebMar 1, 2024 · The profinite group Gis said to have finite rank at most rif each closed subgroup of Gcan be topologically generated by at most relements. A somewhat less obvious result of the same nature is that a profinite group is covered by countably many procyclic subgroups if and only if it is finite-by-procyclic (see Proposition 2.12in the next … fiberglass boat hull repairs shopsderby college bricklayingWebMar 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... derby college art coursesWebApr 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 fiberglass boat hull repair kits