Penultimate term of lower central series is abelian in nilpotent group of class at least three

Statement
If $$G$$ is a fact about::nilpotent group and the fact about::nilpotence class of $$G$$ is an integer $$m \ge 3$$, then the $$(m-1)^{th}$$ term of the fact about::lower central series of $$G$$ is Abelian. In particular, it is an Abelian characteristic subgroup of $$G$$.

Corollaries

 * Finite nilpotent and every Abelian characteristic subgroup is central implies class at most two
 * Nilpotence class is strictly greater than solvable length in finite nilpotent group of class at least three

Breakdown of corresponding statement for upper central series
The second term of the upper central series of a finite nilpotent group need not be abelian.

Facts used

 * 1) uses::Second half of lower central series of nilpotent group comprises Abelian groups: If $$G$$ is a group of nilpotence class $$m$$, and $$k \ge (m + 1)/2$$, the member $$G_k$$ of the lower central series of $$G$$ is Abelian.

Proof
The proof follows directly from fact (1), and the observation that for $$m \ge 3$$, $$m - 1 \ge (m + 1)/2$$.

Textbook references

 * , Page 214, Exercise 3, Chapter 5