James Wiegold

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”.

Growth sequences of finite groups

During his investigation of the possible non-Hopf kernels for finitely generated groups in [1], Dey proves that the minimum number of generators d ( G n ) of the n -th direct power G n of a non-trival finite group G tends to infinity with n . This has prompted me to ask the question: what are the ways in which the sequence { d ( G n )} can tend to infinity? Let us call this the growth sequence for G ; it is evidently monotone non-decreasing, and is at least logarithmic (Theorem 2.1). This paper is devoted to a proof that, broadly speaking, there are two different types of behaviour. If G has non-trivial abelian images (the imperfect case, § 3), then the growth sequence of G is eventually an arithmetic progression with common difference d(G/G). In special cases (Theorem 5.2) the initial behaviour can be quite nasty. Our arguments in § 3 are totally elementary. If G has only trivial abelian images (the perfect case,§ 4), then the growth sequence of G is eventually bounded above by a sequence that grows logarithmically. It is a simple consequence of this fact that there are arbitrarily long blocks of positive integers on which the growth sequence takes constant values. This is a characteristic property of perfect groups, and indeed it was this feature in the growth sequences of large alternating groups (which I found by using ad hoc permutational arguments) that attracted me to the problem in the first place. The discussion of the perfect case rests on the lovely paper of Hall [2], which was brought to my notice by M. D. Atkinson.

Groups with Boundedly Finite Classes of Conjugate Elements

An n-BFC group is one in which no element has more than n conjugates, and some element has exactly n conjugates. A bound in terms of n for the order of the derived group of an n-BFC group is determined, and it is shown that the derived group of a p-BFC group is cyclic of order p, where p is prime. The case of 4-BFC groups is also considered.

Locally graded groups with all subgroups normal-by-finite

In a paper published in this journal [1], J. T. Buckely, J. C. Lennox, B. H. Neumann and the authors considered the class of CF-groups, that G such that | H : Core G ( H )| is finite for all subgroups H . It is shown that locally finite CF-groups are abelian-by-finite and BCF, that is, there is an integer n such that | H : Core G ( H )| ≤ n for all subgroups H . The present paper studies these properties in the class of locally graded groups, the main result being that locally graded BCF-groups are abelian-by-finite. Whether locally graded CF-groups are BFC remains an open question. In this direction, the following problems is posed. Does there exist a finitely generated infinite periodic residually finite group in which all subgroups are finite or of finite index? Such groups are locally graded and CF but not BCF.

LOCALLY FINITE GROUPS ALL OF WHOSE SUBGROUPS ARE BOUNDEDLY FINITE OVER THEIR CORES

For n a positive integer, a group G is called core-n if H/HG has order at most n for every subgroup H of G (where HG is the normal core of H, the largest normal subgroup of G contained in H). It is proved that a locally finite core-n group G has an abelian subgroup whose index in G is bounded in terms of n. 1991 Mathematics Subject Classification 20D15, 20D60, 20F30.

Generators for Alternating and Symmetric Groups

Let Γ denote the modular group, that is, the free product of a group of order 2 and a group of order 3. Morris Newman investigates in [2] the factor-groups of Γ and calls them Γ-groups for short; thus a group is a Γ-group if and only if it has a generating set consisting of an element of order dividing 2 and an element of order dividing 3. Newmans interest centres on finite simple Γ-groups. He proves that the linear fractional groups LF (2, p ) for primes p are Γ -groups, and poses the problem of deciding which of the alternating groups enjoy this property.

GROUPS WITH MANY PERMUTABLE SUBGROUPS

We consider the influence on a group G of the condition that every infinite set of cyclic subgroups of G contains a pair that permute and prove (Theorem 1) that finitely generated soluble groups with this condition are centre-by-finite, and (Theorem 2) that torsion free groups satisfying the condition are abelian.

Growth sequences of finite semigroups

The growth sequence of a finite semigroup S is the sequence { d(S n ) }, where S n is the n th direct power of S and d stands for minimum generating number. When S has an identity, d(S n ) = d(T n ) + kn for all n , where T is the group of units and k is the minimum number of generators of S mod T . Thus d(S n ) is essentially known since d(T n ) is (see reference 4), and indeed d(S n ) is then eventually piecewise linear. On the other hand, if S has no identity, there exists a real number c > 1 such that d(S n ) ≥ c n for all n ≥ 2.

Two-generator groups II

Let n be an odd integer greater than 9. It is proved that the alternating group A n has spread 3 in the sense that for any non-trivial elements x 1 , x 2 , x 3 of A n , there is an element y in A n such that 〈 x i , y 〉 = A n for i = 1, 2, 3.

Commutator subgroups of finite p-groups

Within five minutes of the start of my first meeting with Bernhard Neumann as his research student, late in 1954, he suggested the following problem to me. Let G be a group in which the cardinals of the classes of conjugate elements are boundedly finite with maximum η, say. Then the commutator subgroup G ′ is finite [6], Is the order | G ′| of G ′ bounded in terms of n ? I distinctly recall these words of Neumann: “That should provide us with a start, I think”. He was right: more than just a start, the problem has been a continuing stimulus to a study of questions in fields as far apart as permutation group theory ([7], [11], [12], [14], and some unpublished work of Peter M. Neumann) and multiplicator theory ([13], [2]), as well as attracting interest in its own right.