On difficult problems and locally graded groups

Macedońska, O.

doi:10.1007/s10958-007-0102-9

Cited by 5 publications

(5 citation statements)

References 22 publications

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“…We will now quote a useful result, due to Longobardi, Maj, Smith [7], that provides a sufficient condition for a quotient to be locally graded. Remind the reader that, by Zelmanov's solution of the restricted Burnside problem, locally graded groups of finite exponent are locally finite (see for example [8,Theorem 1]).…”

Section: Proofs Of the Main Resultsmentioning

confidence: 99%

On Groups Admitting a Word Whose Values Are Engel

Bastos

Shumyatsky

Tortora

et al. 2013

Int. J. Algebra Comput.

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Let m, n be positive integers, v a multilinear commutator word and w = v m . We prove that if G is a residually finite group in which all wvalues are n-Engel, then the verbal subgroup w(G) is locally nilpotent. We also examine the question whether this is true in the case where G is locally graded rather than residually finite. We answer the question affirmatively in the case where m = 1. Moreover, we show that if u is a non-commutator word and G is a locally graded group in which all u-values are n-Engel, then the verbal subgroup u(G) is locally nilpotent. then x is a (left) n-Engel element. A group G is called n-Engel if all elements of G are n-Engel. It is a long-standing problem whether any n-Engel group is locally nilpotent. Following Zelmanov's solution of the restricted Burnside problem [16,17], Wilson proved that this is true if G is residually finite [14]. Later the second author showed that if in a residually finite group G all commutators [x 1 , . . . , x k ] are n-Engel, then the subgroup [x 1 , . . . , x k ] | x i ∈ G is locally nilpotent [10,11]. This suggests the following conjecture.Conjecture. Let w be a group-word and n a positive integer. Assume that G is a residually finite group in which all w-values are n-Engel. Then the corresponding verbal subgroup w(G) is locally nilpotent. Preliminary resultsGiven subgroups X and Y of a group G, we denote by X Y the smallest subgroup of G containing X and normalized by Y .Lemma 2.1. Let x and y be elements of a group G satisfying [x, n y m ] = 1, for some n, m ≥ 1. Then x y is finitely generated.Proof. Set X = x y m . Then X is finitely generated by [9, Exercise 12.3.6]. Since x y = X y i | i = 0, . . . , m − 1 , the lemma follows.Corollary 2.2. Let y be an element of a group G and H a finitely generated subgroup. If y m is Engel for some m ≥ 1, then H y is finitely generated.The following lemma is well-known. We supply the proof for the reader's convenience.

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Section: Proofs Of the Main Resultsmentioning

confidence: 99%

On Groups Admitting a Word Whose Values Are Engel

Bastos

Shumyatsky

Tortora

et al. 2013

Int. J. Algebra Comput.

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“…The following result is a consequence of Restricted Burnside Problem (see for instance [11,Theorem 1]). We are now in a position to prove Theorem A.…”

Section: Proofs Of the Main Resultsmentioning

confidence: 99%

On Residually Finite Groups with Engel-like Conditions

Bastos

2016

Communications in Algebra

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Let m, n be positive integers. Suppose that G is a residually finite group in which for every element x ∈ G there exists a positive integer q = q(x) m such that x q is n-Engel. We show that G is locally virtually nilpotent. Further, let w be a multilinear commutator and G a residually finite group in which for every product of at most 896 w-values x there exists a positive integer q = q(x) dividing m such that x q is n-Engel. Then w(G) is locally virtually nilpotent.2010 Mathematics Subject Classification. 20F45; 20E26.

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“…According to Lemma 3.2, G/H is locally graded. By Zelmanov's solution of the Restricted Burnside Problem [15,17,18], locally graded groups of finite exponent are locally finite (see for example [8,Theorem 1]), and so G/H is finite. Thus H is finitely generated and so it is nilpotent.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

On residually finite groups satisfying an Engel type identity

Silveira

2020

Monatsh Math

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Let n, q be positive integers. We show that if G is a finitely generated residually finite group satisfying the identity [x, n y q ] ≡ 1, then there exists a function f (n) such that G has a nilpotent subgroup of finite index of class at most f (n). We also extend this result to locally graded groups.Theorem 1.1. Let G be a finitely generated residually finite group satisfying the identity [x, n y q ] ≡ 1. Then there exists a function f (n)

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On difficult problems and locally graded groups

Abstract: Some problems which have a negative answer in general, have an affirmative answer in the class of locally graded groups and a negative answer outside of this class. We present three of such problems and mention another three, which possibly are of that type.

Cited by 5 publications

References 22 publications

On Groups Admitting a Word Whose Values Are Engel

On Groups Admitting a Word Whose Values Are Engel

On Residually Finite Groups with Engel-like Conditions

On residually finite groups satisfying an Engel type identity

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