Lower central series

Definition

Symbol-free definition

The lower central series of a group is a descending chain of subgroups, indexed by ordinals, where:

• The member indexed by a successor ordinal is the commutator subgroup between its predecessor and the whole group.
• The member indexed by a limit ordinal is the intersection of all its predecessors.

Definition with symbols

Let $G$ be a group. The lower central series of $G$ is indexed by the ordinals as follows:

• When $\alpha =\beta +1$ is a successor ordinal, then $G_{\alpha }=[G,G_{\beta }]$.
• When $\alpha$ is a limit ordinal, $G_{\alpha }=$ ∪ $G_{\gamma }$ varying over $\gamma <\alpha$.

Property theory

Member-wise property theory

Each ordinal gives a subgroup-defining function, namely the ordinal $\alpha$ gives the function sending $G$ to $G_{\alpha }$. $G_{1}$ is the commutator subgroup.

By virtue of each member being a functionally defined subgroup, it is characteristic. Howveer, the particular way in which we have made the definitions in fact tells us that each of these members is a verbal subgroup.

Related group properties

If there is a finite ordinal $c$ for which $G_{c}$ is trivial, then $G$ is nilpotent with nilpotence class $c$.

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.