Some Algebraic and Topological Properties of the Nonabelian Tensor Product

Otera, Daniele Ettore; Russo, Francesco G.; Tanasi, Corrado

doi:10.4134/bkms.2013.50.4.1069

Cited by 8 publications

(11 citation statements)

References 18 publications

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“…Further results (see [1,17]) show that η(G, H) is isomorphic to the free product with amalgamation on a suitable subgroup of G and H. The following results hold. Proposition 5.1 shows that we may connect the notion of nonabelian tensor product with that of free product, because of (5.1).…”

Section: Open Problemsmentioning

confidence: 65%

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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: Open Problemsmentioning

confidence: 65%

“…Therefore, restricting the structure of the nonabelian tensor product π 2 (A, C)⊗π 2 (B, C), we get information on π 3 (X, A, B). This is very interesting in algebraic topology (see [1,16,17]). …”

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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“…where ∧ denotes the operator of nonabelian exterior square in [4,15,16,19] and the last equality is proved in [13,Lemma 2.2]. For instance, d ∧ (E) ≤ d(E) (see [13]) so the commutativity and the exterior degrees are connected.…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

“…This section collects a series of information and some technical results in [4,10,11,16,19]. We begin to recall that a pro-p-group G = lim N ∈P(G) G/N may be written as the projective limit of finite p-groups, where…”

Section: Preliminariesmentioning

confidence: 99%

Commuting Elements With Respect to the Operator Λ in Infinite Groups

Rezaei¹,

Russo²

2016

Bulletin of the Korean Mathematical Society

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Using the notion of complete nonabelian exterior square G ∧G of a pro-p-group G (p prime), we develop the theory of the exterior degree d(G) in the infinite case, focusing on its relations with the probability of commuting pairs d(G). Among the main results of this paper, we describe upper and lower bounds for d(G) with respect to d(G). Here the size of the second homology group H 2 (G, Zp) (over the p-adic integers Zp) plays a fundamental role. A further result of homological nature is placed at the end, in order to emphasize the influence of H 2 (G, Zp) both on G and d(G).

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“…All the groups of the present paper are supposed to be finite. Having in mind the exponential notation for the conjugation of two elements x and y in a group G, that is, the notation x y = y −1 xy, we may follow [3,4,13] in saying that two normal subgroups H and K of G act compatibly upon each other, if for all h 1 , h 2 ∈ H and k 1 , k 2 ∈ K, and if H and K act upon themselves by conjugation. Given h ∈ H and k ∈ K, the nonabelian tensor product H ⊗ K is the group generated by the symbols h ⊗ k satisfying the relations h 1 h 2 ⊗ k 1 = (h h1 2 ⊗ k 1 h1 ) (h 1 ⊗ k 1 ) and…”

Section: The Relative Tensor Degreementioning

confidence: 99%