A long time ago Ju. E. Vapne ([2], [3]) and, independently, the author ([4], [5]) classified those standard and complete wreath products that have faithful representations of finite degree over (commutative) fields. See [6] pages 37 40 & 150–154 for an account of this. Recently, in connection with finitary linear groups, I needed a more general wreath product. Somewhat to my surprise neither the classification nor the proof for these generalized wreath products was a straightforward translation from the standard case. The situation is intrinsically more complex and it seems worthwhile recording it separately.

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.

Wielandt [4] has shown that a common subnormal subgroup of two permutable subgroups of finite group is subnormal in their product. When G is infinite it seems unlikely that Wielandt's theorem will still be true, but an example illustrating this appears to be difficult, even if G is an FC-group (that is groups in which each element has only finitely many conjugates; see [2]). However, if we replace subnormality by ascendancy we have the following.

We study the residual finiteness of free products with amalgamations and HNN-extensions of finitely generated nilpotent groups. We give a characterization in terms of certain conditions satisfied by the associated subgroups. In particular the residual finiteness of these groups implies the possibility of extending the isomorphism of the associated subgroups to an isomorphism of their isolated closures in suitable overgroups of the factors (or the base group in case of HNN-extensions).

Groups in which every infinite set of subgroups contains a pair that permute were studied by M. Curzio, J. Lennox, A. Rhemtulla and J. Wiegold. The question whether periodic groups in this class were locally finite was left open. Here we show that if the generators of such a group G are periodic then G is locally finite. This enables us to get the following characterisation. A finitely generated group G is centre-by-finite if and only if every infinite set of subgroups of G contains a pair that permute.

In Theorem 7.13 of [1], Proposition 3.1 of [5], and Theorem 1 of [10], minimal group actions on R-trees are considered. If a group G acts on a tree T, then a Lyndon length function lu is associated with each point u∈T. Abstract minimal length functions are defined in Section 2 of this paper by a simple reduction process, where lengths of elements are reduced by a fixed amount (except that any length must remain non-negative). It is shown in Theorem 2.3 that minimal length functions correspond to minimal actions by following Chiswell's construction of actions on trees from length functions, given in [4]. A parallel result to Theorem 1 of [10] is given for minimal length functions in Theorem 2.2. One outcome of these results is that to determine which length functions can arise from an action of a group on the same tree, it suffices to consider only minimal length functions. Section 1 is concerned with some preparatory properties on lengths of products of elements. These lead in Proposition 1.6 to an alternative description of the maximal trivializable subgroup associated with a length function, defined in [3].

Here we mean growth in the sense of Milnor and Gromov. After a brief survey of known results, we compute the growth series of the groups , with respect to generators {x, y}. This is done using minimal normal forms obtained by informal use of judiciously chosen rewrite rules. In both of these examples the growth series is a rational function, and we suspect that this is not the case for the Baumslag-Solitar group

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In 1980, McCaughan and Stonehewer showed that a finite soluble group in which every subnormal subgroup has defect at most two has derived length at most nine and Fitting length at most five, and gave an example of derived length five and Fitting length four. In 1984 Casolo showed that derived length five and Fitting length four are best possible bounds.

In this paper we show that for groups of odd order the bounds can be improved. A group of odd order with every subnormal subgroup of defect at most two has derived and Fitting length at most three, and these bounds are best possible.

Let G be a finitely generated group and let R be a commutative ring, regarded as a G-module with G acting trivially. We shall determine when the cup product of two elements of H1(G, R) is zero. Our method will use the interpretation of H2(G, R) as extensions of G by R. This will give an alternative demonstration of results of Hillman and Würfel.

A Neumann subgroup of the classical modular group is by definition a complement of a maximal parabolic subgroup. Recently Neumann subgroups have been studied in a series of papers by Brenner and Lyndon. There is a natural extension of the notion of a Neumann subgroup in the context of any finitely generated Fuchsian group Γ acting on the hyperbolic plane H such that Γ/H is homeomorphic to an open disk. Using a new geometric method we extend the work of Brenner and Lyndon in this more general context.

A group G is said to be conjugacy p-separable if two non-conjugate elements of G remain non-conjugate in some finite p-group endomorphic image of G. We show that the non-cyclic free centre-by-metabelian groups are not conjugacy p-separable for any prime p. On the other hand, we show that every free centre-by-metabelian group has the solvable conjugacy problem

In this paper we continue our investigations of a construction method for subnear-rings of M(G) proposed by H. Wielandt. For a meromorphic product H, H ⊂ Gk, G finite, we obtain necessary and sufficient conditions for M(G, k, H) to be a near-field.

We determine the structure of a nonabelian group G of odd order such that some automorphism of G sends exactly (1/p)|G| elements to their cubes, where p is the smallest prime dividing |G|. These groups are close to being abelian in the sense that they either have nilpotency class 2 or have an abelian subgroup of index p.

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.

A group G is called normally (subnormally) detectable if the only normal (subnormal) subgroups in any direct product G1 × … × Gn of copies of G are just the direct factors Gi. We give an internal characterization of finite subnormally detectable groups and obtain analogous results for associative rings and for Lie algebras. The main part of the paper deals with a study of normally detectable groups, where we verify a conjecture of T. O. Hawkes in a number of special cases.

In this paper the question is considered of when the wreath product of a nilpotent group with a CLT group G is a CLT group. It is shown that if the field with Pr elements is a splitting field of a Hall P1–subgroup of G, then P wr G is a CLT group for all p–groups P with |P/P1|≥ pr. Moreover, the class of all groups G having the property that N wr G is a CLT group for every nilpotent group N is shown to be quite large. For exmple, every group of odd order can be embedded as a subgroup of a group belonging to this class.

It is shown that if {Gn: n = 1, 2,…} is a countable family of Hausdorff kω-topological groups with a common closed subgroup A, then the topological amalgamated free product *AGn exists and is a Hausdorff kω-topological group with each Gn as a closed subgroup. A consequence is the theorem of La Martin that epimorphisms in the category of kω-topological groups have dense image.

Let G be a finite group and let Aut(G) be its automorphism group. Then G is called a k-orbit group if G has k orbits (equivalence classes) under the action of Aut(G). (For g, hG, we have g ~ h if ga = h for some Aut(G).) It is shown that if G is a k-orbit group, then kGp + 1, where p is the least prime dividing the order of G. The 3-orbit groups which are not of prime-power order are classified. It is shown that A5 is the only insoluble 4-orbit group, and a structure theorem is proved about soluble 4-orbit groups.

Let G be a group acting faithfully on a homogeneous tree of order p + 1, p > 1. Let be the space of functions on the Poission boundary ω, of zero mean on ω. When p is a prime. G is a discrete subgroup of PGL2(Qp) of finite covolume. The representations of the special series of PGL2(Qp), Which are irreducible and unitary in an appropriate completion of , are shown to be reducible when restricted to G. It is proved that these representations of G are algebraically reducible on and topologically irreducible on endowed with the week topology.