If G is a group such that every infinite subset of G contains a commuting pair of elements then G is centre-by-finite. This result is due to B. H. Neumann. From this it can be shown that if G is infinite and such that for every pair X, Y of infinite subsets of G there is some x in X and some y in Y that commute, then G is abelian. It is natural to ask if results of this type would hold with other properties replacing commutativity. It may well be that group axioms are restrictive enough to provide meaningful affirmative results for most of the properties. We prove the following result which is of similar nature.
If G is a group such that for each positive integer n and for every n infinite subset X1,...,Xn of G there exist elements xi of Xi
i = 1,... ,n, such that the subgroup generated by {x1
,... ,xn
} is finite, then G is locally finite.