The lower central series of a group G is the sequence of subgroups Gn (n > 1) of G defined inductively by (i) G1 = G, (ii) Gn+1 = [G,Gn], where, for any two subgroups H,K of G, [H,K] denotes the subgroup of G generated by the commutators [h,k] = h-1k-1hk with h Є H, k Є K. The subgroups Gn also satisfy (iii) Gn+1 ⊆ Gn, (iv) [Gn,Gn] ⊆ Gm+n.

Let Ln(G) = Gn/Gn+1, with the operation in this abelian group denoted additively, and let in : Gn + Ln = Ln(G) be the canonical surjection. Then the graded abelian group L(G) = ⊕n > 1 Ln(G) has a natural Lie algebra structure over ℤ where the bracket of two homogeneous elements ∈ = im(x), n = in(y) is defined by [∈,n] = im+n([x,y]) (cf. [8]).

If x ∈ Gn, x ∉ Gn+1, then n = ω(x) is called the weight of x and ∈ = in(x) is called the initial form of x. If x ∈ Gn for n > 1, then the initial form of x is defined to be zero.