This is the first of three papers (the others by the first author alone) which determine all varieties of nilpotent groups of class (at most) four. The initial step is to reduce the problem to two cases: varieties whose free groups have no elements of order 2, and varieties whose free groups have no nontrivial elements of odd order. The varieties of the first kind form a distributive lattice with respect to order by inclusion (which is not a sublattice in the lattice of all group varieties). We give an embedding of this lattice in the direct product of six copies of the lattice which consist of 0 (as largest element) and the odd positive integers ordered by divisibility. The six integer parameters so associated with a variety directly match a (finite) defining set of laws for the variety. We also show that the varieties of the second kind do form a sublattice in the lattice of all varieties. That (nondistributive) sublattice will be treated, in a similarly conclusive manner, in the subsequent papers of this series.

In this paper we complete the investigation of those varieties of nilpotent groups of class (at most) four whose free groups have no nontrivial elements of odd order. Each such variety is labelled by a vector of sixteen parameters, each parameter a nonnegative integer or ∞, subject to numerous but simple conditions. Each vector satisfying these conditions is in fact used and directly yields a defining set of laws for the variety it labels. Moreover, one can easily recognise from the parameters whether one variety is contained in another. In view of the reduction carried out in the first paper of this series (written jointly with L. G. Kovács) this completes the determination of all varieties of nilpotent groups of class four.

The first paper (written jointly with L. G. Kovács) of this three-part series reduced the problem of determining all varieties of the title to the study of the varieties of nilpotent groups of class (at most) four whose free groups have no nontrivial elements of odd order. The present paper deals with these under the additional assumption that the variety contains all nilpotent groups of class three. We label each such variety by a vector of eleven parameters, each parameter a nonnegative integer or ∞, subject to numerous but simple conditions. Each vector satisfying these conditions is in fact used, and matches directly a (finite) defining set of laws for the variety it labels. Moreover, one can readily recognize from the parameters whether one variety is contained in another. The third paper will complete the determination of all varieties of nilpotent groups of class four.

The Fitting class (of finite, soluble, groups), , is said to be Hall π-closed (where π is a set of primes) if whenever G is a group in and H is a Hall π-subgroup of G, then H belongs to . In this paper, we study the Hall π-closure of products of Fitting classes. Our main result is a characterisation of the Hall π-closedFitting classes of the form (where denotes the so-called smallest normal Fitting class), subject to a restriction connecting π with the characteristic of . We also characterise those Fitting classes (respectively, ) such that (respectively, ) is Hall π-closed for all Fitting classes . In each case, part of the proof uses a concrete group construction. As a bonus, one of these construction also yields a “cancellation result” for certain products of Fitting classes.

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”.

The two problem, both raised in the literature, are: (I) Is there, amongst all the permutational products (p.p.s.) on the amalgam = (A, B; H) at least one which is a minimal generalized regular product? (II) If one of the p.p.s. on is isomorphic to the generalized free product (g.f.p.) F on U are they all? We answer both of them negatively.

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 group G is called semi-n-abelian, if for every g ∈ G there exists at least one a(g) ∈ G-which depends only on g-such that (gh)n = a-1(g)gnhna(g) for all h ∈ G; a group G is called n-abelian, if a(g) = e for all g ∈ G. According to Durbin the following holds for n-abelian groups: If G is n-abelian for at lesast 3 consecutive integers, then G in n-abelian for all integers and these groups are exactly the abelian groups. In this paper this problem is generalized to the semi-n-abelian case: If a finite group G is semi-n-abelian for at least 4 consecutive integers then G is semi-n-abelian for all integers and these groups are exactly the nilpotent groups, where the Sylow-2-subgroup is abelian, the Sylow-3-subgroup is any element of the Levi-variety ([[g, h], h] = e ∀ g, h ∈ G) and the Sylow-p-subgroup (p < 3) is of class <2. As a consequence we get a description of all finite (3-)groups, which are elements of the Levi-variety.

It is shown that if m, n are relatively prime positive integers, then the variety consisting of those soluble groups of exponent mn in which any subgroup of exponent m or n is abelian has a basis of two-variable laws.

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.

The completions of certain nilpotent groups with respect to some ascending sequences of integral domains are constructed. These completions are generalizations of Lazard completions for the groups under consideration and they are Lie algebras over the first integral domain in the sequence. The construction is possible in particular for finite p-groups of exponent p and class < p.

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.

If is a saturated formation of finite soluble groups and G is a finite group whose -residual A is abelian then it is well known that G splits over A and the complements are conjugate. Hartley and Tomkinson (1975) considered the special case of this result in which is the class of nilpotent groups and obtained similar results for abelian-by-hypercentral groups with rank restrictions on the abelian normal subgroup. Here we consider the super-soluble case, obtaining corresponding results for abelian-by-hypercyclic groups.