Upper central series not is strongly central

Statement

The Upper central series (?) of a Nilpotent group (?) need not be a Strongly central series (?).

Definitions used

Upper central series

Further information: Upper central series

Strongly central series

Further information: 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. 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 $c\geq 3$, 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 $2c-2\geq c+1$ for $c\geq 3$. Thus, $Z_{2}(G)$ is Abelian.

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