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.