Representation of Group Elements as Short Products

We prove that every group G of order n has t,log n/log 2+0dog log n) elements x, x, such that every group element is a product of the form x,'-• X,'% e ; s 1,0, 11. The result is true more generally for quasigroups. As a corollary we obtain that for n even, every one-factorization of the complete graph on n vertices contains at most r one-factors whose union is connected. The aim of this note is to solve two problems by one theorem. The first problem is related to the title of the paper, the second to one-factorizations of the complete graph. Let G be a finite group of order n. The first problem is : does there exist a small ordered set x t ,. .. . xt of generators of G such that every group element occurs as a subproduct is sufficient. Erdős and Rényi [3] (see also [6, Vol. 3 ., pp. 319-329 .]) studied a similar problem for abelian groups. They proved that for an abelian group G and a random choice of group elements x t ,. .. , x, the probability that every element of G can be represented in the form (1) goes to 1 with n-* x. provided t>~ log n + log log n +w (3) log 2 log 2 where a) " tends to infinity arbitrarily slowly. We don't know the answer to the same problem for non-abelian groups. Problem 1. Does there exist a constant c such that for an arbitrary group G of order n and a random choice of elements x 1 ,. .. , x r of G, the probability that every member of G is represented in the form (1) tends to 1 while n , oo and t>clogn? 27 xi'-• x ',' whereE i e{0,1 1 f Clearly t > log n/log 2. On the other hand, we prove that (1) t >1 log + logo log n 2 + (2) 2 g g