lowercentral

Lower central series of a group G is a descending sequence of subgroups defined by γ1(G) = G and γi+1(G) = [γi(G), G], where [A,B] denotes the subgroup generated by all commutators [a,b] = a^{-1}b^{-1}ab with a ∈ A, b ∈ B. Each γi is normal in G, and γi+1 ≤ γi for all i.

The group G is nilpotent of class c if γc+1(G) = {e}. Equivalently, the quotient G/γc+1(G) is nilpotent of class at most c. If the series reaches the identity after finitely many steps, G is nilpotent; otherwise it is not.

Examples: If G is abelian, γ2(G) = {e} and γi(G) = {e} for i ≥ 2, so G has class 1. The Heisenberg group, the group of 3×3 upper triangular matrices with ones on the diagonal, has γ2 equal to its center and γ3 = {e}, giving class 2.

Relation to other concepts: the lower central series is different from the derived series, which is defined by G(0) = G and G(i+1) = [G(i), G(i)]. The lower central series measures how far commutators with the whole group fail to vanish at successive weights.

Associated graded: gr(G) = ⊕i γi/γi+1 is a graded Lie algebra under a bracket induced by the group commutator. This construction is used in the study of nilpotent groups and in Lie-theoretic approaches to group theory.

Applications: analysis of nilpotent and p-groups, structure theory, and algorithmic computations in combinatorial and computational group theory.

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