Lower central series

Definition
The lower central series of a group is a descending series of subgroups indexed by the ordinals. For a group $$G$$ and an ordinal $$\alpha > 0$$, the $$\alpha^{th}$$ member of the lower central series is denoted $$\gamma_\alpha(G)$$. A more ambiguous notation that may nonetheless be used in some situations for the $$\alpha^{th}$$ member is $$G_\alpha$$.

By default, the term is used to refer to only the finite part of the series, i.e. the series $$G_n$$, for $$n \in \mathbb{N}$$. This looks like:

$$G_1 = G \ge G_2 = [G,G] \ge G_3 = G,G],G] = [G,[G,G \ge G_4 = [G,G],G],G] = [G,[G,[G,G] = [G,G,G],G = G,[G,G,G] \ge \dots$$

For infinite ordinals, we have:

$$G_\omega = \bigcap_{n \in \mathbb{N}} G_n, G_{\omega + 1} = [G_\omega,G] = [G,G_\omega], G_{\omega + 2} = G_\omega,G],G] = [G,[G,G_\omega$$

For a nilpotent group
For a nilpotent group, the lower central series terminates in finitely many steps at the trivial subgroup, and if $$G_{c+1}$$ is the first member which is trivial, then $$G$$ is said to have nilpotency class $$c$$. For a nilpotent group, the lower central series is the fastest descending central series, i.e., if we have a central series:

$$G = H_1 \ge H_2 \dots H_n = \{ e \}$$

Then each $$H_i \ge G_i$$, and thus, $$n \ge c + 1$$.

Subgroup properties satisfied by members
Each ordinal gives a subgroup-defining function, namely the ordinal $$\alpha$$ gives the function sending $$G$$ to $$G_{\alpha}$$. $$G_1$$ is the whole group, while $$G_2$$ is the derived subgroup (also called the commutator subgroup).

By virtue of each member arising from a subgroup-defining function, it is characteristic. Further, the particular way in which we have made the definitions in fact tells us that all the $$G_\alpha$$ for finite $$\alpha$$ are verbal subgroups, while all the $$G_\alpha$$ (even for infinite $$\alpha$$ are fully invariant).

There must exist a (possibly infinite) ordinal $$\alpha$$ such that $$G_{\alpha + 1} = G_\alpha$$. The subgroup $$G_\alpha$$ is called the hypocenter of $$G$$.

Related group properties
If there is a finite ordinal $$c$$ for which $$G_{c+1}$$ is trivial, then $$G$$ is nilpotent with nilpotency class $$c$$. The smallest such $$c$$ is termed the nilpotency class of $$G$$.

If $$G_{\omega}$$ is trivial where $$\omega$$ denotes the first infinite ordinal, then the group is termed residually nilpotent.

If for some infinite ordinal $$\alpha$$, $$G_{\alpha}$$ is the trivial group, then $$G$$ is termed hypocentral.

Relation with upper central series
For a nilpotent group, the lower central series and upper central series are closely related. They both have the same length, and there is a containment relation between them, which follows from the combination of the facts that upper central series is fastest ascending central series and lower central series is fastest descending central series. However, they need not coincide. Nilpotent groups where they do coincide are termed UL-equivalent groups, and nilpotent not implies UL-equivalent.

Here is a table with some distinctions/contrasts between the two central series:

Relation with derived series
The derived series of a group is a series where each member is defined as the derived subgroup of its predecessor, and the zeroth member is the whole group. The lower central series and derived series are related as follows: the $$k^{th}$$ member of the derived series is contained in the $$2^k$$-th member of the lower central series. This follows from the fact that lower central series is strongly central.