A characterization of infinite 3-abelian groups

Abdollahi, Aliréza

doi:10.1007/s000130050373

Cited by 7 publications

(5 citation statements)

References 11 publications

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“…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%

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: Introductionmentioning

confidence: 99%

A combinatorial condition on a certain variety of groups

Taeri

2001

Arch. Math.

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“…As an immediate consequence of the answer of B. H Neumman to the question of P. Erdős [9], we have F ∪ V(w) = V(w * ), where w(x, y) = [x, y]. Further questions of similar nature, with slightly different aspects, have been considered by many authors (see, for example, [1], [2], [3], [4], [13], [11], [12], [7], [8]).…”

Section: Introduction and Resultsmentioning

confidence: 91%

Criteria for laws between infinite subsets of infinite groups

Boukaroura¹,

Bouchlaghem²

2010

Publ. Math. Debrecen

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A theorem of B. H. Neumman shows that infinite group in which every two infinite subsets there exist two commuting elements, is abelian.In this paper, we prove that if in an infinite group G, every two infinite subsets X and Y , there exist a ∈ X and bMoreover, and using this result, we also prove that an infinite group satisfies the law (x n 1 1 x n 2 2 . . . x nr r ) 2 = 1 if and only if in any r infinite subsets X 1 , . . . , X r , of G there exist ai ∈ Xi(i = 1, . . . , r) such that (a n 1 1 . . . a nr r ) 2 = 1, where n1, . . . , nr ∈ {2 k /k ∈ N * } and r ≥ 2.

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“…It is natural to ask: Is every infinite virtually f-ring an f-ring? The similar question for groups has been studied by many people (see for example [1,7,11]). The definition of virtually f-rings and formulation of the above question, as far as we know, first appeared in [2], where a virtually f-ring has been called an f # -ring.…”

Section: Introduction and Resultsmentioning

confidence: 98%

Rings virtually satisfying a polynomial identity

Abdollahi

Akbari

2005

Journal of Pure and Applied Algebra

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Let R be a ring and f (x 1 , . . . , x n ) be a polynomial in noncommutative indeterminates x 1 , . . . , x n with coefficients from Z and zero constant. The ring R is said to be an f-ring if f (r 1 , . . . , r n ) = 0 for all r 1 , . . . , r n of R and a virtually f-ring if for every n infinite subsets X 1 , . . . , X n (not necessarily distinct) of R, there exist n elements r 1 ∈ X 1 , . . . , r n ∈ X n such that f (r 1 , . . . , r n ) = 0. Let R * be the 'smallest' ring (in some sense) with identity containing R such that Char(R) = Char(R * ). Then denote by Z R the subring generated by the identity of R * and denote byf R the image of f in Z R [x 1 , . . . , x n ] (the ring of polynomials with coefficients in Z R in commutative indeterminates x 1 , . . . , x n ). In this paper, we show that if R is a left primitive virtually f-ring such thatf R = 0, then R is finite. Using this result, we prove that an infinite semisimple virtually f-ring R is an f-ring, if the subring of Z R generated by the coefficients off R is equal to Z R ; and we also prove that if f (where ∈ {−1, 1}, then every infinite virtually f-ring with identity is a commutative f-ring. Finally we show that a commutative Noetherian virtually f-ring R with identity is finite if the subring generated by the coefficients off R is Z R .

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A characterization of infinite 3-abelian groups

Abstract: In this note we prove that every infinite group G is 3-abelian (i.e. ab 3 a 3 b 3 for all aY b in G) if and only if in every two infinite subsets X and Y of G there exist x P X and y P Y such that xy 3 x 3 y 3 .

Cited by 7 publications

References 11 publications

A combinatorial condition on a certain variety of groups

A combinatorial condition on a certain variety of groups

Criteria for laws between infinite subsets of infinite groups

Rings virtually satisfying a polynomial identity

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