Lower central series

Definition

Symbol-free definition

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

• The first member is the group itself
• 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 (there are two notations: $\gamma _{\alpha }(G)$ is the more unambiguous notation, while some also use $G_{\alpha }$):

• $\gamma _{1}(G)=G_{1}=G$
• When $\alpha =\beta +1$ is a successor ordinal, then $\gamma _{\alpha }(G)=G_{\alpha }=[G,G_{\beta }]$.
• When $\alpha$ is a limit ordinal, $G_{\alpha }=\bigcap G_{\beta }$ (or, $\gamma _{\alpha }(G)=\bigcap _{\beta <\alpha }\gamma _{\beta }(G)$)

Often, 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:

$G_{1}=G\geq G_{2}=[G,G]\geq G_{3}=[[G,G],G]\geq \dots$

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 nilpotence class $c$. For a nilpotent group, the lower central series is the fastest descending central series, i.e., if we have a central series:

$G=H_{1}\geq H_{2}\dots H_{n}=\{e\}$

Then each $H_{i}\geq G_{i}$, and thus, $n\geq c+1$.

Further information: Lower central series is fastest descending central series

Facts

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 commutator subgroup.

By virtue of each member arising from a subgroup-defining function, it is characteristic. Howveer, 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 characteristic).

Related group properties

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