Sheila Oates Macdonald

On the orbit-sizes of permutation groups containing elements separating finite subsets

It is proved that if G is a permutation group on a set Ω every orbit of which contains more than mn elements, then any pair of subsets of Ω containing m and n elements respectively can be separated by an element of G.

On Conway's conjecture for integer sets

Let A = {a i } be a finite set of integers and let p, m denote the orders of A + A = { a i + a j } and A – A = { a i – a j } respectively. J.H. Conway conjectured that p ≤ m always and that p = m only if A is symmetric about 0. This conjecture has since been disproved; here we make several other observations on the values of p and m .

Laws in finite strictly simple loops

It is shown that a finite loop with no proper nontrivial subloops has a finite basis for its laws.

On the laws of certain linear groups

In recent years a great deal of attention has been devoted to the study of finite nonabelian simple groups, but one aspect which seems to have been little considered is that of the laws which they satisfy. In a recent paper, two of the present authors gave a basis for the laws of PSL(2, 5) (Cossey and Macdonald [ l ]) , and here we present bases for the laws of PSL(2, 7) and PSL(2, 9). An important feature of the proofs of these results is the knowledge of a two variable basis for the laws of 54 (the symmetric group of degree 4), which we also state. We show how bases for certain SL(2, p) can be derived from bases for the corresponding PSL(2f p). Finally, we give some laws which hold in PSL(2, p) (p a prime) in more general cases. In notation and terminology we follow the book of Hanna Neumann [2]. We would draw the readers attention particularly to the law vn and its properties, given in 52.31 and 52.32 of [2]. Also note that var G denotes the variety generated by a group G. Our results are as follows.

Locally finite varieties of groups arising from Cross varieties

Let [formula omitted] be a Cross variety and let n be the least integer such that [formula omitted] is locally finite; then n ≤ 2d + 3 where d is an upper bound for the number of generators of certain critical groups in [formula omitted].

Locally finite varieties of groups arising from Cross varieties: Corrigendum

One condition on D was omitted from the statement of Theorem B in the paper [1], namely that (and the corresponding condition in Theorem A). The proof of the theorem clearly requires that this be so and it is not difficult to see that such a D can always be chosen. However, Dr R.M. Bryant has pointed out that not only is the theorem true as it stands, but indeed the conditions can be relaxed slightly to allow D to be any group such that .