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, $G_{\alpha }=\bigcap _{\beta <\alpha }G_{\beta }$ (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:

$G_{1}=G\geq G_{2}=[G,G]\geq G_{3}=[[G,G],G]\geq G_{4}=[[[G,G],G],G]\geq \dots$

For infinite ordinals, we have:

$G_{\omega }=\bigcap _{n\in \mathbb {N} }G_{n}$.

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:

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

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.

Subgroup series properties

Strongly central series

This subgroup series-defining function yields a strongly central series.

The lower central series of a group is a strongly central series. In other words, if $m,n$ are natural numbers, then $[G_{m},G_{n}]\leq G_{m+n}$. For full proof, refer: Lower central series is strongly central

This has some important consequences. For instance: second half of lower central series of nilpotent group comprises Abelian groups, nilpotent and every Abelian characteristic subgroup is central implies class at most two, solvable length is logarithmically bounded by nilpotence class.

Strongly characteristic series

This subgroup series-defining function does not yield a strongly characteristic series.

The lower central series of a group is not a strongly characteristic series. In other words, it is not necessary that a smaller member of the lower central series is a characteristic subgroup in a bigger member. This is despite the fact that all members are characteristic subgroups, and in fact are verbal subgroups, in the whole group. For full proof, refer: Lower central series not is strongly characteristic