Upper central series not is strongly central

Statement
The fact about::upper central series of a fact about::nilpotent group need not be a fact about::strongly central series.

Related facts

 * Lower central series is strongly central
 * Nilpotent not implies UL-equivalent: In a nilpotent group, the upper and lower central series need not be the same.
 * Upper central series may be tight with respect to nilpotence class

Facts used

 * 1) uses::Upper central series may be tight with respect to nilpotence class: For any natural number $$c$$, we can construct a nilpotent group $$G$$ such that if $$Z_k(G)$$ denotes the $$k^{th}$$ member of the upper central series, the part of the upper central series upto $$Z_k(G)$$ is also the upper central series of $$Z_k(G)$$. In particular, $$Z_k(G)$$ has nilpotence class $$k$$.

Proof
For $$c \ge 3$$, consider a group $$G$$ that fits the situation of fact (1). Then, $$Z_2(G)$$ has class exactly equal to two.

Suppose now that the upper central series of $$G$$ were strongly central. Then, when numbered from $$G$$ downwards, $$Z_2(G)$$ is the $$(c-1)^{th}$$ member, so by the definition of strongly central, $$[Z_2(G),Z_2(G)]$$ is in the $$(2c-2)^{th}$$ member from $$G$$ downwards, which is the trivial subgroup since $$2c - 2 \ge c + 1$$ for $$c \ge 3$$. Thus, $$Z_2(G)$$ is Abelian.

This is a contradiction, so the upper central series of $$G$$ is not strongly central.