A characterisation of cyclic subnormal separated A-groups of nilpotent length three

The paper gives a detailed description of all those finite A-groups of nilpotent length three that satisfy the cyclic subnormal separation condition. It is shown that every monolithic group under discussion is an extension of its Fitting subgroup P, which is a homocyclic p-group, by a p′ metabelian subgroup H, where p is a prime. The centraliser of P in H is trivial while the monolith W is equal to ω1(P) and the action of H on W is faithful and irreducible. H is further shown to have non trivial centre and is an extension of its derived subgroup M by a subgroup L such thatfor all primes q where Mq and Lq are the respective Sylow q-subgroups of M and L. The Fitting subgroup of F of H is shown to be M × Z(H), while Z(H) = F ∩ L and every element of L of prime order is in Z(H). Finally it is shown that if ql(q) is the exponent of Mq then every element of order dividing ql(q) in L belongs to Z(H).

McLain groups over arbitrary rings and orderings

In 1954, McLain [M] applied some well-known arguments of linear algebra on triangular matrices to establish the existence of characteristically simple, locally finitep-groups, now known as McLain groups [Rob, pp. 347–349]. His groups, having a trivial centre, illustrated sharply the difference between finite and locally finitep-groups. The construction of McLain groups depends on the dense linear ordering (ℚ,≤) and a fieldFpofpelements. It was immediately clear that the parametersFp, ℚ of the McLain groupG(Fp, ℚ) could be replaced by other linearly ordered, dense setsSand by other fieldsFwithout doing much harm to the construction. IfFhas characteristic 0, thenG(F, S) is still locally nilpotent but torsion-free. Wilson [W] investigatedG(Fp, S) for other orderings and Roseblade[R] deeply studied the automorphism group AutG(Fp, S) in his dissertation at Cambridge in 1963.

The general theory of 3-transposition groups

A set D of 3-transpositions in the group G is a normal set of elements of order 2 such that, for all d and e in D, the order of the product de is 1, 2, or 3. If G is generated by the conjugacy class D of 3-transpositions, we say that (G, D) is a 3-transposition group or (loosely) that G is a 3-transposition group. The study of 3-transposition groups was instituted by Bernd Fischer [6, 7, 8] who classified all 3-transposition groups which are finite and have no non-trivial normal solvable subgroups. Recently the present author and H. Cuypers[5] extended Fischer's result to include all 3-transposition groups with trivial centre. For this classification the present paper provides the extension of Fischer's paper [8] where he gave two basic reductions, the Normal Subgroup Theorem and the Transitivity Theorem stated below. Other results of help in the classification are also presented here.

A class of one-relator groups with centre

Let the group H have presentation where m ≥ 3, pi ≥ 2 and (pi, pj) = 1 if i ≠ j. We show that H is a one-relator group precisely if H can be obtained from a suitable group 〈a, b; ap = bp〉 by repeated applications of a (two-stage) procedure consisting of applying central Nielsen transformations followed by adjoining a root of a generator. We conjecture that any one-relator group G with non-trivial centre and G/G′ not free abelian of rank two can be obtained in the same way from a suitable group 〈a, b; ap = bp〉.

On the centre and residual finiteness of the automorphism group of a group ring

Let G be a group and let Aut(G) be its automorphism group. It is notorious that the properties of Aut (G) do not relate well to the properties of G, perhaps the only twogeneral results being that if G has a trivial centre then the same is true of Aut (G) [2, p.89] and Baumslag's theorem that if G is finitely generated and residually finite then Aut (G) is also residually finite [1, Theorem 1, p. 117]. In the paper we shall attempt tofind analogues of these results for therelationship between the properties of R(G), the group ring of G over a ring R, and the properties of Aut R(G), the automorphism of R(G). We prove that if R(G) has a trivial centre then Aut R(G) has a trivial centre. We establish the analogue, Theorem 2.3, of Baumslag's theorem by ring-theoretic methods; our original proof used properties of group rings, the present simplified proof we owe to the referee. As an example we calculate Aut ℤ(G) in the case that G is the direct product of two cyclic groups, one of infinite order and the other of order 5. This calculation will, it is hoped, give some indication of the difficulties in determining automorphisms of the group ring of an infinite group.

Torsion in semicomplete nilpotent groups

Let Aut G and Inn G denote the group of all automorphisms of the group G and the subgroup of all inner automorphisms of G, respectively. A group G is said to be complete if it has trivial centre and Aut G = Inn G. Examples of such groups abound and they have been the object of study for many years. Following Heineken (8) we call a group G semicomplete if Aut G = Inn G.