Centre-by-metabelian groups with the maximal condition for normal subgroups are exhibited which (a) are residually finite but have quotient groups which are not residually finite; and (b) have all quotients residually finite but are not abelian-by-polycyclic.

We describe the structure of the lower central factors of free centre-by-metabelian groups.

This is an investigation of whether a group epimorphism maps the maximal perfect subgroup of its domain onto that of its image. It is shown how the question arises naturally from considerations of algebraic K-theory and Quillen's plus-construction. Some sufficient conditions are obtained; these relate to the upper central series, or alternatively the derived series, of the domain. By means of topological/homological techniques, the results are then sharpened to provide, in certain circumstances, conditions which are necessary as well as sufficient.

We prove a conjecture of Lennox and Wiegold that a finitely generated soluble group, in which every infinite subset contains two elements generating a supersoluble group, is finite-by-supersoluble.

In this note a formation U is considered which can be defined by a sequence of laws which ‘almost’ hold in every finite supersoluble group. The class U contains all finite supersoluble groups and each group in U has a Sylow tower.

It is shown that a finite group belongs to U if and only if all of its subgroups with nilpotent commutator subgroup are supersoluble. A more general result concerning classes of this type finally proves that U is a saturated formation.

A finite variety is a class of finite groups closed under taking subgroups, factor groups and finite direct products. To each such class there exists a sequence w1, w2,… of words such that the finite group G belongs to the class if and only if wk(G) = 1 for almost all k. As an illustration of the theory we shall present sequences of words for the finite variety of groups whose Sylow p-subgroups have class c for c = 1 and c = 2.

The main results are as follows. A finitely generated soluble group G is polycyclic if and only if every infinite set of elements of G contains a pair generating a polycyclic subgroup; and the same result with “polycyclic” replaced by “coherent”.

In 1957 P. Hall conjectured that every (finitely based) variety has the property that, for every group G, if the marginal factor-group is finite, then the verbal subgroup is also finite. The content of this paper is to present a precise bound for the order of the verbal subgroup of a G when the marginal factor-group is of order Pn (p a prime and n > 1) with respect to the variety of polynilpotent groups of a given class row. We also construct an example to show that the bound is attained and furthermore, we obtain a bound for the order of the Baer-invariant of a finite p-group with respect to the variety of polynilpotent groups.

A classical result of M. Zorn states that a finite group is nilpotent if and only if it satisfies an Engel condition. If this is the case, it satisfies almost all Engel conditions. We shall give a similar description of the class of p-soluble groups of p-length one by a sequence of commutator identities.

If G, H and B are groups such that G × B ≃ H × B, G/[G, G]. Z(G) is free abelian and B is finitely generated abelian, then G ≃ H. The equivalence classes of triples (Vξ,A) where Vand A are finitely generated free abelian groups and ξ: V⊗ V → A is a bilinear form constitute a semigroup B undera natural external orthogonal sum. This semigroup B is cancellative. A cancellation theorem for class 2 nilpotent groups is deduced.

A group G is termed conjugacy separable (c.s.) if any pair of distinct conjugacy classes may be mapped to distinct conjugacy classes in some finite epimorph of G. The free product of A and B with cyclic amalgamated subgroup H is shown to be c.s. if A and B are both free, or are both finitely generated nilpotent groups. Further, one-relator groups with nontrivial center and HNN extensions with c.s. base group and finite associated subgroups are also c.s.

Let G/G' be finitely generated and let G = B1 x A1 = B2 x A2 = … = Bi x Ai = … with each Bi isomorphic to a fixed group B which obeys the maximal condition for normal subgroups. Then the Ai represent only finitely many isomorphism classes. We give an example with B infinite cyclic, G/G' free abelian of infinite (countable) rank and such that G is decomposed as above with no two Ai isomorphic.

Lyndon's axiomatic methods are used in [1] to show, among other things, that a group G with an integer valued length function satisfying certain conditions is free. At the end of his paper [2] Lyndon gives a method of embedding such a group in a free group whose natural length function extends the function on G. We construct here a simpler embedding with the same property.

Ore (1942) studied the automorphisms of finite monomial groups and Holmes (1956, pp. 23–93) has given the form of the automorphisms of the restricted monomial groups in the infinite case. The automorphism group of a standard wreath product has been studied by Houghton (1962) and Segal (1973, Chapter 4). Monomial groups and standard wreath products are both special cases of permutational wreath product. Here we investigate the automorphisms of the permutational wreath product and consider to what extent the results holding in the special cases remain true for the general construction. Our results extend those of Bunt (1968).

A length function, for a group, associates to an element x a real number |x| subject to certain axioms, including a cancellation axiom which embodies certain cancellation properties for elements of a free group. Integer valued length functions were introduced by Roger Lyndon [1] where, with a more restrictive set of axioms than ours, it is shown that a length function for a group is given by a restriction of the usual length function on some free product.