John C. Lennox

Extensions of a problem of Paul Erdös on groups

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”.

Bigenetic properties of finitely generated hyper- (Abelian-by-finite) groups

Let be a class and p a property of groups. We say that p is a bigenetic property of p-groups (or more simply, p is bigenetic in p-groups) if an p-group G has the property p whenever all two-generator subgroups of G have p.

The subnormal embedding of complete groups

In [ 1) Dark records a result of Philip Hall that the symmetric group S, of degree three cannot be embedded subnormally in a finite perfect group, that is, a group which is equal to its own derived subgroup. Now S, is the smallest complete group, that is, a group with trivial centre and no outer automorphisms and our objective here is to show that Hall’s result is a specific case of a more general phenomenon. In fact we have

CYCLICALLY SEPARATED GROUPS

We call a group G cyclically separated if for any given cyclic subgroup B in G and subgroup A of finite index in B , there exists a normal subgroup N of G of finite index such that N ∩ B = A . This is equivalent to saying that for each element x ∈ G and integer n ≥ 1 dividing the order o ( x ) of x , there exists a normal subgroup N of G of finite index such that Nx has order n in G/N . As usual, if x has infinite order then all integers n ≥ 1 are considered to divide o ( x ). Cyclically separated groups, which are termed “potent groups” by some authors, form a natural subclass of residually finite groups and finite cyclically separated groups also form an interesting class whose structure we are able to describe reasonably well. Construction of finite soluble cyclically separated groups is given explicitly. In the discussion of infinite soluble cyclically separated groups we meet the interesting class of Fitting isolated groups, which is considered in some detail. A soluble group G of finite rank is Fitting isolated if, whenever H = K/L ( L ⊲ K ≤ G ) is a torsion-free section of G and F ( H ) is the Fitting subgroup of H then H / F ( H ) is torsion-free abelian. Every torsion-free soluble group of finite rank contains a Fitting isolated subgroup of finite index.

On a centrality property of finitely generated torsion free soluble groups

We recall from [5] that a group G is centrally eremitic, or merely eremitic, if there exists a positive integer e such that whenever an element of G has some power in a centralizer, it has its e-th power. The eccentricity of an eremitic group G is the least such positive integer e. In [5] it was proved among other results that finitely generated Abehan-by-nilpotent groups are centrally eremitic. In [6] Wehrfritz showed that finitely generated linear groups over fields of characteristic zero are centrally eremitic and possess a subgroup of finite index with eccentricity 1. He asked whether finiteIy generated Abelian-by-nilpotent groups also have the latter property. This paper is devoted to answering this question. In the negative direction we show first of all that there exists a twogenerator metabehan group which does not have a subgroup of finite index with eccentricity 1. Let G be the standard wreath product L wr iIs of a cyclic group L of order 2 with an infinite cyclic group M. Let A be the base group of G and suppose G* is a subgroup of G with eccentricity 1. Then G* n A is of exponent 2 and so it is trivial that the square of any element of G* n d centralizes every element of G*. Hence G* n 3 is contained in the centre of G*, since G* has eccentricity 1. Moreover G*/G* n d is isomorphic to G*d,/A, an Abelian group. Therefore GU is nilpotent. It is clear that G has no nilpotent subgroups of finite index, and the result follows. The above argument rests on the fact that the base group of G is periodic and so the question naturally arises as to what happens in a finitely generated group G which has a torsion free Abelian normal subgroup d with G/,q nilpotent. As our main result we shall prove

An analogue for semigroups of a group problem of P. Erdös and B. H. Neumann

In response to a question posed by P. Erdos, B. H. Neumann showed that in a group with every subset of pairwise noncommuting elements finite there is a bound on the size of these sets. Recently, H. E. Bell, A. A. Klein and the first author showed that a similar result holds for rings. However in the case of semigroups, finiteness of subsets of pairwise noncommuting elements does not assure the existence of a bound for their size. The largest class of semigroups in which we found Neumanns result valid are cancellative semigroups.

A note on orthogonality and permutability of subnormal subgroups

Let H and K be groups with derived sub groups H´ and K´ . Following (5) we say that H and K are orthogonal, and write H ⊥ K , if and only if the tensor product H / H´ ® K / K´ is trivial. If H and K are sub groups of a group we say that H and K permute if and only if HK = KH. We recall finally that the subgroup H of the group G is said to be subnormal in G , written H sn G , if and only if there exists a finite chain of subgroups connecting H to G .

Lower Frattini series in finitely generated soluble groups

It is proved that some term of the lower Frattini series of a finitely generated abelian-by-polycyclic group is trivial. The analogous result for finitely generated centre-by-metabelian groups is false.

Some remarks on coherent soluble groups

An example of Wehrfritz is pointed out to show that GL(4, Q ) is not coherent. This answers a question of Serre. It is shown that finitely generated soluble coherent subgroups of GL(2, Q ) need not be polycyclic, in sharp contrast to the fact that all soluble subgroups of GL( n, Z ) are polycyclic and so automatically coherent.