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.

A presentation is given for the cohomology ring of a finitely presented combinatorially aspherical group with trivial coefficients in an integral domain. Cohomological periodicity is characterized in terms of the cup product.

A construction for Fitting formations given by the author and C. L. Kanes is generalised. The original examples were based on the use of Fitting families of modules over algebraically closed fields. An example of Haberl and Heineken in 1984 suggested that the methods should work with modules over arbitrary fields. We show that this is indeed the case, provided we restrictthe class of groups considered.

In order to classify solvable groups Philip Hall introduced in 1939 the concept of isoclinism. Subsequently he defined a more general notion called isologism. This is so to speak isoclinism with respect to a certain variety of groups. The equivalence relation isologism partitions the class of all groups into families. The present paper is concerned with the internal structure of these families.

Certain central products of the binary polyhedral groups with finite cyclic groups are here shown to have presentations with two generators and two defining relations; this disproves a conjecture of the second author, stated in J. Austral. Math. Soc. Ser. A 38 (1985), 230–240.

This is a study of formal power series under the binary operation of formal composition from a group-theoretical point of view. Various “large” properties are derived.

We give new presentations of the five Mathieu groups, the simple groups J1, J2, HS, McL, Co3, and some other simple and related groups. All generators in these presentations are involutions. Our presentations are simpler than the known presentations of this type for the groups mentioned above.

Let G be a group factorized by finitely many pairwise permutable nilpotent subgroups. The aim of this paper is to find conditions under which at least one of the factors is contained in a proper normal subgroup of G.

A method for constructing Fitting-Schunck classes is given: the method is an adaptation of one given by C. L. Kanes for constructing Fitting formations, and generalizes the Fitting-Schunck class construction given by Cossey in 1981. A criterion for deciding which of the Fitting-Schunck classes so constructed are formations is given.