Two-Generator conditions for Residually Finite Groups

Wilson, John S.

doi:10.1112/blms/23.3.239

Cited by 72 publications

(47 citation statements)

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“…However it is unknown whether or not n-Engel groups, that is, those satisfying the law [x, n y] = 1 for some fixed n, must be locally nilpotent (although this seems unlikely). This has been established for n < 3 (see [6]), and, for general n, for the class of residually finite n-Engel groups [11]. (Note that there are relatively easy examples of non-nilpotent «-Engel groups; see, for example, [8, p. 132] In the present note we call attention to a simple general fact about Engel groups which has apparently hitherto gone unnoticed, and from it infer firstly the local nilpotence of Engel 'SB-groups' (these are defined below; they include soluble groups), and then a quite specific global description of the n-Engel groups in a large class "io of groups (including soluble and residually finite groups), yielding in particular their local nilpotence.…”

Section: [[Gh]h]h] = Lmentioning

confidence: 82%

A note on Engel groups and local nilpotence

Burns¹,

Medvedev

1998

J Aust Math Soc A

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This paper is concerned with the question of whether n-Engel groups are locally nilpotent. Although this seems unlikely in general, it is shown here that it is the case for the groups in a large class <& including all residually soluble and residually finite groups (in fact all groups considered in traditional textbooks on group theory). This follows from the main result that there exist integers c(n), e(n) depending only on n, such that every finitely generated n-Engel group in the class <€ is both finite-of-exponent-*?(n)-bynilpotent-of-class< c(n) and nilpotent-of-class< c(«)-by-finite-of-exponent-e(n). Crucial in the proof is the fact that a finitely generated Engel group has finitely generated commutator subgroup.

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Section: [[Gh]h]h] = Lmentioning

confidence: 82%

A note on Engel groups and local nilpotence

Burns¹,

Medvedev

1998

J Aust Math Soc A

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“…The first is due to Zel'manov [27] and the second to J. Wilson [25]. For a short proof of the latter see [4].…”

Section: Theorem Z2 Every N-engel Lie Algebra Is Locally Nilpotentmentioning

confidence: 99%

On locally finite 𝑝-groups satisfying an Engel condition

Abdollahi

Traustason²

2002

Proc. Amer. Math. Soc.

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Abstract. For a given positive integer n and a given prime number p, let r = r(n, p) be the integer satisfying p r−1 < n ≤ p r . We show that every locally finite p-group, satisfying the n-Engel identity, is (nilpotent of n-bounded class)-by-(finite exponent) where the best upper bound for the exponent is either p r or p r−1 if p is odd. When p = 2 the best upper bound is p r−1 , p r or p r+1 . In the second part of the paper we focus our attention on 4-Engel groups. With the aid of the results of the first part we show that every 4-Engel 3-group is soluble and the derived length is bounded by some constant.

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“…In a variety in which every polycyclic group is nilpotent, each finitely generated residually finite group is nilpotent. This result may be considered as an extension of a theorem of Wilson [17], stating that each n-Engel finitely generated residually finite group is nilpotent. In short, we have 2 ) i C l C U,>o ?…”

Section: Corollary 3 Leff Be a Variety Of Groups In Which Every Polymentioning

confidence: 75%