In a paper published in this journal [1], J. T. Buckely, J. C. Lennox, B. H. Neumann and the authors considered the class of CF-groups, that G such that |H: CoreG (H)| is finite for all subgroups H. It is shown that locally finite CF-groups are abelian-by-finite and BCF, that is, there is an integer n such that |H: CoreG(H)| ≤ n for all subgroups H. The present paper studies these properties in the class of locally graded groups, the main result being that locally graded BCF-groups are abelian-by-finite. Whether locally graded CF-groups are BFC remains an open question. In this direction, the following problems is posed. Does there exist a finitely generated infinite periodic residually finite group in which all subgroups are finite or of finite index? Such groups are locally graded and CF but not BCF.

A group variety defined by one semigroup law in two variables is constructed and it is proved that its free group is not a periodic extension of a locally soluble group.

A group G has all of its subgroups normal-by-finite if H / coreG(H) is finite for all subgroups H of G. These groups can be quite complicated in general, as is seen from the so-called Tarski groups. However, the locally finite groups of this type are shown to be abelian-by-finite; and they are then boundedly core-finite, that is to say, there is a bound depending on G only for the indices | H: coreG(H)|.

The aim of this paper is to consider Problem 1 posed by Stewart and Wiegold in [6]. The main result is that if G is a finitely generated perfect group having non-trivial finite images, then there exists a finite image B of G such that the growth sequence of B is eventuallly faster than that of every finite image of G. Moreover we investigate the growth sequences of simple groups of the same order.

We give an alternative short proof of a recent theorem of J. A. Hillman and P.A. Linnell that an elementary amenable group with finite Hirsch number has, modulo its locally finite radical, a soluble normal subgroup with index and derived length bounded only in terms of the Hirsch number of the group.

We consider (finite) groups in which every two-generator subgroup has cyclic commutator subgroup. Among other things, these groups are metabelian modulo their hypercentres, and in the corresponding quotient group all subgroups of the commutator subgroup are normal.

Regular maps of type {p, q}r and the associated groups Gp,q,r are considered for small values for p, q and r. In particular, it is shown that the groups G4,6,6 and G5,5,6 are Abelian-by-infinite, and there are infinitely many regular maps of each of the types {4, 6}6, {5, 5}6, {5, 6}6 and {6, 6}6.

Any representation of a group G on a vector space V extends uniquely to a representation of G on the free metabelian Lie algebra on V. In this paper we study such representations and make some group-theoretic applications.

Examples of Fitting classes generated by certain periodic infinite groups are presented in this paper. The groups in question are finite extensions of direct products of quasi-cyclic (Prüfer) groups and are, in particular, Černikov groups. The methods applied follow closely those used for constructing Fitting classes of finite groups, especially in the case of nilpotent length three.

In this paper we consider the groups G = G(α, n) defined by the presentations . We derive a formula for [G′: ″] and determine the order of G whenever n ≦ 7. We show that G is a finite soluble group if n is odd, but that G can be infinite when n is even, n ≧ 8. We also show that G(6, 10) is a finite insoluble group involving PSU(3, 4), and that the group H with presentation is a finite group of deficiency zero of order at least 114,967,210,176,000.

We answer some questions which arise from a recent paper of Campbell, Heggie, Robertson and Thomas on one-relator free products of two cyclic groups. In the process we show how publicly accessible computer programs can be used to help answer questions about finite group presentations.

In this paper we study groups with Černikov conjugacy classes which are nilpotent-by-Černikov groups, giving full characterizations of them and applying the results obtained to some related areas.

If G is an elementary amenable group of finite Hirsch length h, then the quotient of G by its maximal locally finite normal subgroup has a maximal solvable normal subgroup, of derived length and index bounded in terms of h.

Our set-up will consist of the following: (i) a graph with vertex set V and edge set E; (ii) for each vertex ∈ V a non-trivial group Gv given by a presentation (xν; rν); (iii) for each edge e = {u, ν} ∈ E a group Ge given by a presentation (xu, xv; re) where re consists of the elements of ru ∪ rv, together with some further words on xu ∪ xv. We let G = (x; r) where x = ∪v∈v xv, r = ∪e∈E re. Ouraim is to try to describe the structure of G in terms of the groups Gv, (v ∈ V), Ge (e ∈ E). Under suitable conditions the natural homomorphisms Gv, → G (ν ∈ V), Ge → Ge (e ε E) are injective; and there is a short exact sequence (where, for any group H, IH is the augmentation ideal). Some (co)homological consequences of these resultsare derived.

We extend earlier work relating asphericity and Euler characteristics for finite complexes whose fundamental groups have nontrivial torsion free abelian normal subgroups. In particular a finitely presentable group which has a nontrivial elementary amenable subgroup whose finite subgroups have bounded order and with no nontrivial finite normal subgroup must have deficiency at most 1, and if it has a presentation of deficiency 1 then the corresponding 2-complex is aspherical. Similarly if the fundamental group of a closed 4-manifold with Euler characteristic 0 is virtually torsion free and elementary amenable then it either has 2 ends or is virtually an extension of Z by a subgroup of Q, or the manifold is asphencal and the group is virtually poly- Z of Hirsch length 4.

The aim of this paper is to prove some embedding theorems for groups with Černikov conjugacy classes. Moreover a characterization of periodic central-by-Černikov groups is given.

This paper classifies all finite connected 4- and 5-arc-transitive cubic graphs that contain circuits of length less than or equal to 11, or of length 13, and some of those graphs with circuits of length 12.

The following question is discussed and evidence for and against it is advanced: is it true that if F is an arbitrary finite subgroup of an arbitrary non-linear simple locally finite group G, then CG(F) is infinite? The following points to an affirmative answer.

Theorem A. Let F be an arbitrary finite subgroup of a non-linear simple locally finite group G. Then there exist subgroups D ◃ C ≤ G such that F centralizes C/D, F∩C ≤ D, and C/D is a direct product of finite alternating groups of unbounded orders. In particular, F centralizes an infinite section of G.

Theorem A is deduced from a “local” version, namely

Theorem B. There exists an integer valued function f(n, r) with the following properties. Let H be a finite group of order at most n, and suppose that H ≤ S, where S is either an alternating group of degree at least f = f(n, r) or a finite simple classical group whose natural projective representation has degree at least f. Then there exist subgroups D ◃ C ≤ S such that (i) [H, C] ≤ D, (ii) H ∩ C ≤ D, (iii) C/D ≅ Alt(r), (iv) D = 1 if S is alternating, and D is a p-group of class at most 2 and exponent dividing p2 if S is a classical group over a field of characteristic p.

The natural “local version” of our main question is however definitely false.

Proposition C. Let p be a given prime. Then there exists a finite group H that can be embedded in infinitely many groups PSL(n, p) as a subgroup with trivial centralizer.

We consider the influence on a group G of the condition that every infinite set of cyclic subgroups of G contains a pair that permute and prove (Theorem 1) that finitely generated soluble groups with this condition are centre-by-finite, and (Theorem 2) that torsion free groups satisfying the condition are abelian.

We give presentations for the groups PSL(2, pn), p prime, which show that the deficiency of these groups is bounded below. In particular, for p = 2 where SL(2, 2n) = PSL(2, 2n), we show that these groups have deficiency greater than or equal to – 2. We give deficiency – 1 presentations for direct products of SL(2, 2n) for coprime ni. Certain new efficient presentations are given for certain cases of the groups considered.