A Natural Linear Ordering of Basic Commutators

The basic commutators derived from the generators of a free group were introduced by Philip Hall and studied extensively by Marshall Hall, Jr. There exists a natural linear ordering for these commutators. It is the purpose of this paper to show that this ordering is, in a certain sense, invariant under "multiplication," i.e., under the process of forming the commutator with the same basic commutator on both sides of an inequality. We derive and state our results in the language of commutators in groups; obviously, they can be formulated also in terms of elements of a free Lie ring. Our investigation was motivated by a study of the smallest normal divisor in a free group containing a given basic commutator. We start out by giving some notation and definitions which will be used throughout this paper. G will be the free group on xi, x2, • • • , xr. If a, bEG then (a, b) = a~1b~1ab. The lower central series of G is the chain dZ)G2Z) • • • GOG+iD • • • of subgroups defined by setting d = G, G„= (Gn-i, G), the group generated by all commutators of the form (an-i, b) with gb-iGG„_i and bEG. We wish now to construct basic commutators and at the same time define a linear ordering on them. This is done by induction. The basic commutators of weight one with their linear order are XiCj and such that if Ci=(C„ Ct), then C^Ct. Let Cni=(Cil, C^) and C„2=(C,2, C,-2) be of weight w. Then Cni> C„2 if Cil > C¿2 or Ctl = C,-, but C,x > Cir A basic commutator of weight w is greater than any of weight less than w. Thus, if r = 2, let Xi = x, x2 = y so that the basic commutators of weight ^ 3 in their order are