The Structure of Compact Groups

Hofmann, Karl Heinrich; Morris, Sidney A.

doi:10.1515/9783110296792

Cited by 141 publications

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“…The proof of these properties can be found in [2,3]. In particular, Brown and others [2,3] show that In order to state our main results, we recall from [6,12,13] that a compact (Hausdorff) group G possessing the filter basis P(G) = {N = N ⊳ G | G/N is a finite p-group} is said to be a pro-p-group (p prime) if G = lim N ∈P(G) G/N, that is, if G is a projective limit of finite p-groups (see [12, Of course, the topology of G is the unique topology induced by P(G). More generally, we may replace P(G) with…”

Section: Statement Of the Main Theoremsmentioning

confidence: 88%

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On the Topology of the Nonabelian Tensor Product of Profinite Groups

Russo¹

2016

Bulletin of the Korean Mathematical Society

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Abstract. The properties of the nonabelian tensor products are interesting in different contexts of algebraic topology and group theory. We prove two theorems, dealing with the nonabelian tensor products of projective limits of finite groups. The first describes their topology. Then we show a result of embedding in the second homology group of a pro-pgroup, via the notion of complete exterior centralizer. We end with some open questions, originating from these two results.

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Section: Statement Of the Main Theoremsmentioning

confidence: 88%

“…and G is said to be a profinite group if G = lim N ∈F (G) G/N, that is, if G is a projective limit of finite groups (see again [6,12,13] for details). A first question is to understand which topology we get on the nonabelian tensor product of profinite groups.…”

Section: Statement Of the Main Theoremsmentioning

confidence: 99%

On the Topology of the Nonabelian Tensor Product of Profinite Groups

Russo¹

2016

Bulletin of the Korean Mathematical Society

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“…https://doi.org/10.1017/S0004972712000573 [7] The probability that x m and y n commute in a compact group 509 direct product G g = G 0 ι Z/nZ where the morphism ι : Z/nZ → Aut G 0 is defined by ι(m + nZ)(h) = g m hg −m . Recall that the multiplication on G g is given by…”

Section: Compact Lie Groups and Measures Respecting Closed Subgroupsmentioning

confidence: 99%

“…Then G lim N∈N(G) G/N (see [7,Lemma 9.1, p. 448]). We shall now keep n 1 and n 2 fixed throughout the remainder of the section and show that Theorem 3.4(1) implies the commutativity of the identity component G o of an arbitrary compact group G provided that all Lie group quotients are n j -straight for j = 1, 2.…”

Section: Consequences For Arbitrary Compact Groupsmentioning

confidence: 99%

“…By Lemma 4.2, G/N is both n 1 -and n 2 -straight. We know It may be useful to recall that G = G 0 D for some profinite group D such that G 0 ∩ D is normal in G. This follows from Dong Hoon Lee's supplement theorem for compact groups (see [7,Theorem 9.41]). However, the issue of commuting powers in the case of a profinite group G is not discussed in this paper.…”

Section: Consequences For Arbitrary Compact Groupsmentioning

confidence: 99%

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The Probability That and Commute in a Compact Group

Hofmann

Russo

2012

Bull. Aust. Math. Soc.

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In a recent article [K. H. Hofmann and F. G. Russo, 'The probability that x and y commute in a compact group', Math. Proc. Cambridge Phil Soc., to appear] we calculated for a compact group G the probability d(G) that two randomly selected elements x, y ∈ G satisfy xy = yx, and we discussed the remarkable consequences on the structure of G which follow from the assumption that d(G) is positive. In this note we consider two natural numbers m and n and the probability d m,n (G) that for two randomly selected elements x, y ∈ G the relation x m y n = y n x m holds. The situation is more complicated whenever n, m > 1. If G is a compact Lie group and if its identity component G 0 is abelian, then it follows readily that d m,n (G) is positive. We show here that the following condition suffices for the converse to hold in an arbitrary compact group G: for any nonopen closed subgroup H of G, the sets {g ∈ G : g k ∈ H} for both k = m and k = n have Haar measure 0. Indeed, we show that if a compact group G satisfies this condition and if d m,n (G) > 0, then the identity component of G is abelian.2010 Mathematics subject classification: primary 20C05, 20P05; secondary 43A05.

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Realizations of rotations on an indecomposable compact monothetic group

Maier

2015

Mathematische Nachrichten

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By the classical Halmos‐von Neumann theorem, each compact monothetic group corresponds to an ergodic dynamical system with discrete spectrum. For such groups we prove two results. We first construct a compact monothetic group which does not split into a direct product of a connected and a totally disconnected compact monothetic group. Then we present a measure preserving dynamical system on the unit square being isomorphic to a rotation on this indecomposable group.

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The Structure of Compact Groups

Cited by 141 publications

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On the Topology of the Nonabelian Tensor Product of Profinite Groups

On the Topology of the Nonabelian Tensor Product of Profinite Groups

The Probability That and Commute in a Compact Group

Realizations of rotations on an indecomposable compact monothetic group

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