On a combinatorial problem in varieties of groups

Endimioni, Gérard

doi:10.1080/00927879508825531

Cited by 11 publications

(16 citation statements)

References 12 publications

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“…If G t is finite, then G is finite-by-nilpotent and therefore residually finite. Hence G ∈ B n , by Corollary 1 in [6]. If G t is infinite, then since G t is locally finite, G t ∈ B n , by Theorem 3 in [6].…”

Section: P R O O F O F T H E T H E O R E M a Let A Be An Infinite Abmentioning

confidence: 94%

“…If the set y B = {y b | b ∈ B} is finite, then |C G (y)| = ∞. So Z(G) is infinite and the result follows from Lemma 3 in [6]. Hence suppose that y B is infinite and consider infinite sets P r o o f. By Lemma 1, every n-Bell group is an Engel group.…”

Section: Lemma 4 Let N ∈ {±2 ±3 4} and G Be An Infinite B * N -Gromentioning

confidence: 96%

“…Now a finitely generated soluble Engel group is nilpotent [7]. Therefore, by Theorem 3 of [6], G ∈ B n , that is G is an n-Bell group.…”

Section: Lemma 4 Let N ∈ {±2 ±3 4} and G Be An Infinite B * N -Gromentioning

confidence: 99%

“…But for some words and some classes of groups the answer is known to be positive (see for example [1], [2], [3], [6], [11], [12], [13], [17], [18]). Let B n be the variety of n-Bell groups defined by the law [x n , y][x, y n ] −1 = 1. n-Bell groups are considered in [4], [9], [10].…”

Section: Introductionmentioning

confidence: 99%

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A combinatorial condition on a certain variety of groups

Taeri

2001

Arch. Math.

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Let n be an integer and B n be the variety defined by the law [x n , y][x, y n ] −1 = 1. Let B * n be the class of groups in which for any infinite subsets X, Y there exist x ∈ X and y ∈ Y such that [x n , y][x, y n ] −1 = 1. For n ∈ {±2, 3} we prove that B * n = B n ∪ F, F being the class of finite groups. Also for n ∈ {−3, 4} and an infinite group G which has finitely many elements of order 2 or 3 we prove that G ∈ B * n if and only if G ∈ B n .

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Section: P R O O F O F T H E T H E O R E M a Let A Be An Infinite Abmentioning

confidence: 94%

Section: Lemma 4 Let N ∈ {±2 ±3 4} and G Be An Infinite B * N -Gromentioning

confidence: 96%

“…Now a finitely generated soluble Engel group is nilpotent [7]. Therefore, by Theorem 3 of [6], G ∈ B n , that is G is an n-Bell group.…”

Section: Lemma 4 Let N ∈ {±2 ±3 4} and G Be An Infinite B * N -Gromentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A combinatorial condition on a certain variety of groups

Taeri

2001

Arch. Math.

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“…In the following theorem we collect some circumstances in which the equality holds. Theorem 1.1 ( [7]). Let V be a variety of groups defined by the law o ¼ 1.…”

Section: Introductionmentioning

confidence: 99%