R. G. Burns

A note on groups with separable finitely generated subgroups

An example is given of an infinite cyclic extension of a free group of finite rank in which not every finitely generated subgroup is finitely separable. This answers negatively the question of Peter Scott as to whether in all finitely generated 3-manifold groups the finitely generated subgroups are finitely separable. In the positive direction it is shown that in knot groups and one-relator groups with centre, the finitely generated normal subgroups are finitely separable.

A note on Engel groups and local nilpotence

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.

On the orbit-sizes of permutation groups containing elements separating finite subsets

It is proved that if G is a permutation group on a set Ω every orbit of which contains more than mn elements, then any pair of subsets of Ω containing m and n elements respectively can be separated by an element of G.

GROUP LAWS IMPLYING VIRTUAL NILPOTENCE

Ifw 1 is a group law implying virtual nilpotence in every finitely generated metabelian group satisfying it, then it implies virtual nilpotence for the finitely generated groups of a large class of groups including all residually or locally soluble-or-finite groups. In fact the groups of satisfying such a law are all nilpotent-by-finite exponent where the nilpotency class and exponent in question are both bounded above in terms of the length ofw alone. This yields a dichotomy for words. Finally, if the law w 1 satisfies a certain additional condition—obtaining in particular for any monoidal or Engel law—then the conclusion extends to the much larger class consisting of all ‘locally graded’ groups. 2000 Mathematics subject classification: primary 20F19, 20E10, 20F45.

Maximal normal subgroups of the integral linear group of countable degree

This paper continues the second authors investigation of the normal structure of the automorphism group г of a free abelian group of countably infinite rank. It is shown firstly that, in contrast with the case of finite degree, for each prime p every linear transformation of the vector space of countably infinite dimension over Z p , the field of p elements, is induced by an element of г Since by a result of Alex Rosenberg GL(א o , Z p ) has a (unique) maximal normal subgroup, it then follows that г has maximal normal subgroups, one for each prime.

Analytic relatively free pro-p groups

Abstract It is shown that a pro-p group which is both relatively free and p-adic analytic must be nilpotent-by-finite, confirming a conjecture of Aner Shalev.

How to Tell if a Group Law Entails Virtual Nilpotence: Postscript

It is known that for any finitely generated group G from the large class of “locally graded” groups, satisfaction of an Engel or positive law forces G to be virtually nilpotent. Black (1999) gives a sufficient condition for an arbitrary 2-variable law w(x, y)≡ 1 to imply virtual nilpotence—though only for finitely generated residually finite groups. We show how the Dichotomy Theorem from Burns and Medvedev (2003) for arbitrary words w(x 1,…,x n ), encompasses Blacks condition, extending it to the n-variable case and a certain large class 𝒮 (however still falling short of the class of locally graded groups). We infer in particular that her condition is also necessary. We also deduce a simplified version of an algorithm of Li (2004, 2005) for deciding whether or not a given law w(x 1,…,x n )≡ 1 satisfies the extended version of Blacks criterion.

Recalcitrance in groups – CORRIGENDUM

The following changes should be made to [1]. The assertion in the proof of Theorem 2 that a group G having solubility length k ‘means simply that commutators of weight k commute with each other’ is incorrect. In fact, groups with this property are just those that are abelian-by-(nilpotent of class at most k − 1). Hence, Theorem 2 should be replaced by the following weaker result. THEOREM 2 (Revised). Let G be a 2-generator Abelian-by-nilpotent group, with commutator quotient G/[G, G] free Abelian of rank 2 (that is, ∼=Z × Z). Then G has recalcitrance ≤3. Everywhere in Section 4 (including the heading) the word ‘soluble’ should thus be changed to ‘Abelian-by-nilpotent’ (and likewise in the abstract and the Introduction). Finally, the sentence in parentheses in the introductory paragraph of Section 4 should be omitted.