On finite groups in which coprime commutators are covered by few cyclic subgroups

Acciarri, Cristina; Shumyatsky, Pavel

doi:10.1016/j.jalgebra.2014.02.033

Cited by 4 publications

(4 citation statements)

References 9 publications

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“…Somewhat surprizingly, it seems that until recently there was no published work on finite groups covered by few cyclic subgroups. That gap was filled in [3]. In particular, the article [3] contains the following results.…”

Section: Coverings By Procyclic Subgroupsmentioning

confidence: 98%

Coverings of commutators in profinite groups

Acciarri¹,

Shumyatsky²

2017

Rend. Sem. Mat. Univ. Padova

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Let w be a group-word. Suppose that the set of all w-values in a profinite group G is contained in a union of countably many subgroups. It is natural to ask in what way the structure of the verbal subgroup w(G) depends on the properties of the covering subgroups. The present article is a survey of recent results related to that question. In particular we survey results on finite and countable coverings of word-values (mostly commutators) by procyclic, abelian, nilpotent, and soluble subgroups, as well as subgroups with finiteness conditions. The last section of the paper is devoted to relation of the described results with Hall's problem on conciseness of group-words.2010 Mathematics Subject Classification. 20E18; 20F14.

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Section: Coverings By Procyclic Subgroupsmentioning

confidence: 98%

Coverings of commutators in profinite groups

Acciarri¹,

Shumyatsky²

2017

Rend. Sem. Mat. Univ. Padova

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“…It is an easy exercise to show that if G is a finite group, then γ ∞ (G) is generated by the commutators [x, y] such that x, y are elements of G having mutually coprime orders. The following theorem was proved in [2].…”

Section: Recall That In a Groupmentioning

confidence: 98%

Procyclic coverings of commutators in profinite groups

2014

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We consider profinite groups in which all commutators are contained in a union of finitely many procyclic subgroups. It is shown that if G is a profinite group in which all commutators are covered by m procyclic subgroups, then G possesses a finite characteristic subgroup M contained in G ′ such that the order of M is m-bounded and G ′ /M is procyclic. If G is a pro-p group such that all commutators in G are covered by m procyclic subgroups, then G ′ is either finite of m-bounded order or procyclic.2010 Mathematics Subject Classification. 20E18; 20F14.

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“…Every element of G is both a γ * 1 -commutator and a δ * 0 -commutator. Now let k ≥ 2 and let X be the set of all elements of G that are powers of γ Coprime commutators have been studied by many authors (for example see [1,2,6,8,10]). In [8] it is established that the nilpotent residual γ ∞ (G) of a finite group G is generated by commutators of primary elements of coprime orders, where a primary element is an element of prime power order.…”

Section: Theorem 4 ([9]mentioning

confidence: 99%

“…Coprime commutators have been studied by many authors (for example see [1,2,6,8,10]). In [8] it is established that the nilpotent residual γ ∞ (G) of a finite group G is generated by commutators of primary elements of coprime orders, where a primary element is an element of prime power order.…”

Section: Introductionmentioning

confidence: 99%

Coprime commutators in finite groups

Monetta

Bastos

2019

Communications in Algebra

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Let G be a finite group and let k ≥ 2. We prove that the coprime subgroup γ * k (G) is nilpotent if and only if |xy| = |x||y| for any γ * k -commutators x, y ∈ G of coprime orders (Theorem A). Moreover, we show that the coprime subgroup δ * k (G) is nilpotent if and only if |ab| = |a||b| for any powers of δ * k -commutators a, b ∈ G of coprime orders (Theorem B).2010 Mathematics Subject Classification. 20D30, 20D25.

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On finite groups in which coprime commutators are covered by few cyclic subgroups

Abstract: Abstract. The coprime commutators γ

Cited by 4 publications

References 9 publications

Coverings of commutators in profinite groups

Coverings of commutators in profinite groups

Procyclic coverings of commutators in profinite groups

Coprime commutators in finite groups

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