On Locally Graded Groups

Kim, Yangkok; Rhemtulla, A. H.

doi:10.1515/9783110908978.189

Cited by 19 publications

(20 citation statements)

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“…Then by ([10] 31.12), there is a finite normal series with cyclic factors H = N 0 N 1 ... N m = N. We know that N 0 is finitely generated and assume, that N i is finitely generated. Since by assumption G does not contain free non-cyclic subsemigroups, and the group N i /N i+1 is cyclic, it follows from ( [5], Lemmas 5 and 1) that N i+1 is finitely generated, which accomplishes the induction, and proves that N is finitely generated. 2 Lemma 2 A finitely generated group, which is finite-by-nilpotent-by-finite, is nilpotent-by-finite.…”

mentioning

confidence: 76%

Positively discriminating groups

Macedońska¹,

Campbell²,

Quick³

et al. 2007

Groups St Andrews 2005

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A group is positively discriminating if any finite subset of positive equations u = v, which are not laws in G, can be simultaneously falsified in G. All known groups, which are not positively discriminating, satisfy positive laws. The problem whether every group without positive laws must be positively discriminating is open. We give an affirmative answer to the problem in the class of locally graded groups.An equation in a group is an expression of the form u = v, where Let V be a finite set of equations. Since the equations need not be cancelled, we can assume that for some n all the equations in V are written on n variables. If there is an n-tuple in a group G, which falsifies each equation in V, we say that V can be simultaneously falsified in the group G. For example, in symmetric group S 3 the set of two equations {xy = y, xy 2 = x} can be simultaneously falsified by pair of elements a, d ∈ S 3 , of orders 2, 3, respectively: ad = d, ad 2 = a. However the set {xy 2 = x, xy 3 = x} can not be simultaneously falsified, because either y 2 or y 3 has the image 1 in S 3 and hence each pair satisfies at least one equation. More examplesThe following binary sets V of non-laws can not be simultaneously falsified. Each pair of elements satisfies some equation in V:We recall that a group G is discriminating (see [10] 17.12, 17.23), if any finite subset V of non-laws in G, can be simultaneously falsified in G. In terms of [1] it means that G discriminates the free group in var G. If we consider only the subsets V of positive equations, or of binary equations, we can speak of positively or binary discriminating groups, respectively.

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mentioning

confidence: 76%

Positively discriminating groups

Macedońska¹,

Campbell²,

Quick³

et al. 2007

Groups St Andrews 2005

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“…It is shown by Kim and Rhemtulla [11], that every locally graded n-Engel group G is locally nilpotent. Then by a result of Burns and Medvedev [6], G is contained in the variety N c(n) B e(n) ∩ B e(n) N c(n) , where c(n) and e(n) depend on n only.…”

Section: Theorem 4 An N-engel Group G Satisfies a Positive Law If G Imentioning

confidence: 99%

On difficult problems and locally graded groups

Macedońska

2007

J Math Sci

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“…This gives a negative answer to Problem 1 in general, however we show in Theorem 2, that in the class of locally graded groups the answer is affirmative. Proof It is shown by Kim and Rhemtulla [5], that every locally graded nEngel group G is locally nilpotent. Then by a result of Burns and Medvedev [10], G is contained in a variety N c B e ∩ B e N c , where c and e depend on n only.…”

Section: Problemsmentioning

confidence: 99%

“…Let H be a two-generator subgroup in G. Since H also is locally graded and collapsing, then by ( [5] Theorem A and Lemma 3), H contains a polycyclic normal subgroup N of finite index. Then by the result of Rosenblatt ([11], 4.12), the finitely generated soluble F-group N must be nilpotent-by-finite.…”

Section: Problemsmentioning

confidence: 99%