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:

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

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:

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

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

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 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:

Nature of fact	Fact for lower central series	Fact for upper central series
Is the series a strongly central series?	Lower central series is strongly central	upper central series not is strongly central (i.e., the upper central series need not always be a strongly central series).
What is the nilpotency class of the members of the series?	Second half of lower central series of nilpotent group comprises abelian groups, Penultimate term of lower central series is abelian in nilpotent group of class at least three	Upper central series may be tight with respect to nilpotency class
Are the members verbal subgroups and/or fully invariant subgroups in the whole group?	Lower central series members are verbal (and since verbal implies fully invariant, they are also fully invariant)	Upper central series members need not be fully invariant (even for a nilpotent group)

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.

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 $[G_m,G_n]\le G_{m+n}$. 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 nilpotency 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