# 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

## 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 $c \ge 3$, 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.