Robert Geoffrey Burns

York University · Department of Mathematics and Statistics

Motivated by the well-known conjecture of J. J. Andrews and M. L. Curtis [Am. Math. Mon. 73, 21-28 (1966; Zbl 0135.04403)], we consider the question of how, in a given n-generator group G, any ordered n-tuple of “annihilators” of G, that is, with normal closure all of G, can be transformed by standard moves into a generating n-tuple. The recalcitra...

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 (19991. Black , S. ( 1999 ). Which words spell “almost nilpotent”? J. Algebra 221 : 475 – 496 . [CrossRef]View all references) gives a sufficient condition for an ar...

Motivated by a well-known conjecture of Andrews and Curtis, we consider the question as to how in a given n-generator group G, a given set of n "annihilators" of G, that is, with normal closure all of G, can be transformed by standard moves into a generating n-tuple. The recalcitrance of G is defined to be the least number of elementary standard mo...

The Fuchsian groups are the discrete subgroups of LF(2, R), the group of all 2 × 2 matrices over the reels with determinant +1. We are interested here in the following group-theoretical property in particular in connection with Fuchsian groups: a group is said to have the finitely generated intersection property if the intersection of every pair of...

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.

If ω ≡ 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...

Motivated by a well-known conjecture of Andrews and Curtis, we consider the question as to how in a given n-generator group G, a given set of n “annihilators” of G, that is, with normal closure all of G, can be transformed by standard moves into a generating n-tuple. The recalcitrance of G is defined to be the least number of elementary standard mo...

It is shown that ifH,Kare any finitely generated subgroups of a free groupFandUis any cyclic subgroup ofF, then any intersectionHg1U∩Kg2Uof double cosets contains only a finite number of double cosets (H∩K)gU, and an explicit upper bound for this number is given in terms of the ranks ofHandKand a generator ofU. This result is then applied to the in...

Let A * B be the free product of groups A, B. We define in the natural way the Kurosh rank (Krk H) of any subgroup H ≤ A * B, essentially as the number of obvious free factors in the decomposition of H as a free product in accordance with the Kurosh subgroup theorem. We establish the following analogue of the Howson-Hanna Neumann formula for free g...

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 C including all residually soluble and residually finite groups (in fact all groups considered in traditional textbooks on group theory). This follow...

We investigate the structure of groups satisfying apositive law, that is, an identity of the formu ≡ v, whereuandvare positive words. The main question here is whether all such groups are nilpotent-by-finite exponent. We answer this question affirmatively for a large class of groups including soluble and residually finite groups, showing that moreo...

In this paper we bring together a number of results on a problem of Jnsson about characterising a variety of groups by means of the normal-subgroup lattices of its member groups.

An element of a free group is called almost primitive in F, if it is primitive in every proper subgroup confining it, though not in F itself. Several examples of almost primitive elements (APEs) are exhibited. The main results concern the behaviour of proper powers w l of certain APEs w in a free group F (and, more generally, in free products of cy...

Without Abstract

It is shown that the natural generalisations of the elementary Nielsen transformations of a free group to the infinite-rank case, furnish generators for the subgroup of “bounded” automorphisms of any relatively free nilpotent group of infinite rank. This settles the nilpotent analogue of a question of D. Solitar concerning the “bounded” automorphis...

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 i...

In this paper two theorems are proved that give a partial answer to a question posed by G. Behrendt and P. Neumann. Firstly, the existence of a group of cardinality 1 with exactly 1 normal subgroups, yet having a subgroup of index 2 with 21 normal subgroups, is consistent with ZFC (the Zermelo-Fraenkel axioms for set theory together with the Axiom...

The first result gives a (modest) improvement of the best general bound known to date for the rank of the intersection U ∩ V of two finite-rank subgroups of a free group F in terms of the ranks of U and V . In the second result it is deduced from that bound that if A is a finite-rank subgroup of F and B < F is non-cyclic, then the index of A ∩ B in...

In this paper two theorems are proved that give a partial answer to a question posed by G. Behrendt and P. Neumann. Firstly, the existence of a group of cardinality 1 with exactly 1 normal subgroups, yet having a subgroup of index 2 with 21 normal subgroups, is consistent with ZFC (the Zermelo-Fraenkel axioms for set theory together with the Axiom...

Elementary algebraic techniques are used to obtain the precise possible indices of torsion-free subgroups of finite index of finitely generated Fuchsian groups (and related groups).

Elementary algebraic techniques are used to obtain the precise possible indices of torsion-free subgroups of finite index of finitely generated Fuchsian groups (and related groups).

An example is given of a descending sequence of free factors of a free group F whose intersection is not a free factor of F.

This paper continues the second author's 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 Zp, the field of p...

A simple proof is given of a result of Hmelevskiï on the solutions of the equation [ x , y ] = [ u , υ ] over a free group for any specified u , υ . To illustrate, the equation is solved explicitly for ( u , υ ) = ( a , b ), ( a ² , b ), ([ a , b ], c ) (where a, b, c freely generate the free group) and thence stabilizers of the corresponding commu...

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 .

Sufficient conditions are found for the free product G G of two groups A A and B B with an amalgamated subgroup U U to have the properties (1) that the intersection of each pair of finitely generated subgroups of G G is again finitely generated, and (2) that every finitely generated subgroup containing a nontrivial subnormal subgroup of G G has fin...

If H is a subgroup of a group G we shall say that G is H-residually finite if for every element g in G , outside H , there is a subgroup of finite index in G , containing H and still avoiding g . (Then, according to the usual definition, G is residually finite if it is E -residually finite, where E is the identity subgroup). Definitions of other te...

Recently A. L. Šmel'kin [14] proved that a product variety1 is generated by a finite group if and only if is nilpotent, is abelian, and the exponents of and are coprime. Alternatively, by the theorem of Oates and Powell [13], we may say that a Cross variety is decomposable if and only if it is of the above form.(Received November 17 1965)(Revised J...