Groups with no Free Subsemigroups

Longobardi, P.; Maj, M.; Rhemtulla, A. H.

doi:10.2307/2154822

Cited by 7 publications

(4 citation statements)

References 4 publications

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“…Hence G/Z k (G) is also finite [11]. D To prove Theorem 3, we need the following key lemma, whose proof is similar to [15,Proposition 5]. , there exists a finite normal subgroup H of G such that G/H is a torsion-free nilpotent group.…”

Section: Lemma 2 Every Torsion-free Nilpotent £ 3 (Oo)-group Belongmentioning

confidence: 99%

Some Engel conditions on infinite subsets of certain groups

Abdollahi¹

2000

Bull. Austral. Math. Soc.

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Let k be a positive integer. We denote by ɛk(∞) the class of all groups in which every infinite subset contains two distinct elements x, y such that [x,k y] = 1. We say that a group G is an -group provided that whenever X, Y are infinite subsets of G, there exists x ∈ X, y ∈ Y such that [x,k y] = 1. Here we prove that:(1) If G is a finitely generated soluble group, then G ∈ ɛ3(∞) if and only if G is finite by a nilpotent group in which every two generator subgroup is nilpotent of class at most 3.(2) If G is a finitely generated metabelian group, then G ∈ ɛk(∞) if and only if G/Zk (G) is finite, where Zk (G) is the (k + 1)-th term of the upper central series of G.(3) If G is a finitely generated soluble ɛk(∞)-group, then there exists a positive integer t depending only on k such that G/Zt (G) is finite.(4) If G is an infinite -group in which every non-trivial finitely generated subgroup has a non-trivial finite quotient, then G is k-Engel. In particular, G is locally nilpotent.

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Section: Lemma 2 Every Torsion-free Nilpotent £ 3 (Oo)-group Belongmentioning

confidence: 99%

Some Engel conditions on infinite subsets of certain groups

Abdollahi¹

2000

Bull. Austral. Math. Soc.

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“…In the first case N is finitely generated for obvious reason. In the second case we apply the following statements from the paper of Longobardi and Rhemtulla [27,Lemmas 1,2]. Lemma 4.3 If G has no free subsemigroups, then for all a, b ∈ G the subgroup a b n , n ∈ Z is finitely generated.…”

Section: Theorem 42 Let G Be a Finitely Generated Group With No Freementioning

confidence: 99%

On the Gap Conjecture concerning group growth

Grigorchuk

2012

Bull. Math. Sci.

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We discuss some new results concerning Gap Conjecture on group growth and present a reduction of it (and its *-version) to several special classes of groups. Namely we show that its validity for the classes of simple groups and residually finite groups will imply the Gap Conjecture in full generality. A similar type reduction holds if the Conjecture is valid for residually polycyclic groups and just-infinite groups. The cases of residually solvable groups and right orderable groups are considered as well.

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“…R2. If G is a finitely generated group without free nonabelian subsemigroups, then all derived subgroups of G are finitely generated (by [8], Corollary 3).…”

Section: Lemma 1 the Class C Is Closed Under Taking Subgroupsmentioning

confidence: 99%

On the smallest locally and residually closed class of groups, containing all finite and all soluble groups

Bajorska¹

2006

Publ. Math. Debrecen

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The class C was introduced in [1] as an infinite union of certain classes of groups. We provide a simpler characterization of the class C. We confirm in the class C the conjecture by R. Grigorchuk, that all groups of intermediate growth are residually finite. We also consider the larger class obtained from C by closing under extentions. We show that Malcev conjecture -that every finitely generated group satisfying a positive law must be nilpotent-by-finite -known to hold in the class C, remains true in the extended class cl C.

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Groups with no Free Subsemigroups

Cited by 7 publications

References 4 publications

Some Engel conditions on infinite subsets of certain groups

Some Engel conditions on infinite subsets of certain groups

On the Gap Conjecture concerning group growth

On the smallest locally and residually closed class of groups, containing all finite and all soluble groups

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