# Lower central series is strongly central

This fact is an application of the following pivotal fact/result/idea:three subgroup lemma

View other applications of three subgroup lemma OR Read a survey article on applying three subgroup lemma

## Contents

## Statement

The Lower central series (?) of a Nilpotent group (?) is a Strongly central series (?).

## Explanation

Intuitively, what we're saying is that the slowest way to make commutators fall is by bracketing them completely to one side. Thus, for instance, doing a bracketing like:

is bigger than the subgroup:

This is closely related to the fact that the property of being a nilpotent group, which is characterized by the lower central series reaching the identity, is substantially stronger than the property of being a solvable group, which is characterized by the derived series reaching the identity.

## Related facts

### Stronger facts

- Centralizer relation between lower and upper central series: This states that members of the lower central series centralize corresponding members of the upper central series.

### Applications

- Second half of lower central series of nilpotent group comprises Abelian groups
- Solvable length is logarithmically bounded by nilpotence class
- Penultimate term of lower central series is Abelian in nilpotent group of class at least three
- Nilpotent and every Abelian characteristic subgroup is central implies class at most two

### Breakdown for upper central series

- Upper central series not is strongly central: There are groups where the upper central series is not a strongly central series.

## Facts used

## Proof

*Given*: A nilpotent group , the lower central series of defined by ,

*To prove*:

*Proof*: We prove the result by induction on (letting vary freely; note that we need to apply the result for multiple values of for the same in the induction step).

**Base case for induction**: For , we have equality:

**Induction step**: Suppose we have, for all , that . Now, consider the three subgroups:

Applying the three subgroup lemma to these yields that is contained in the normal closure of the subgroup generated by and .

We have:

- (by induction assumption)
- (where the first inequality is by induction assumption)

Since is normal, the normal closure of the subgroup generated by both is in , hence the three subgroup lemma yields:

which is what we require.