Upper central series

Definition

Definition with symbols

The upper central series of a group $G$ is an ascending chain of subgroups indexed by ordinals, wher ethe $\alpha ^{th}$ member is denoted as $Z^{\alpha }(G)$. It is defined as follows:

• When $\alpha =\beta +1$ is a successor ordinal, $Z^{\alpha }(G)$ is the inverse image of the center of $G/Z^{\beta }(G)$ with respect to the natural projection map $G$ → $Z^{\beta }(G)$.

• When $\alpha$ is a limit ordinal, $Z^{\alpha }(G)$ is the union of all $Z^{\gamma }(G)$ where $\gamma <\alpha$.

The zeroth member is the trivial group and the first member is the center.

Property theory

Member-wise property theory

Each ordinal $\alpha$ gives a subgroup-defining function that sends $G$ to $Z^{\alpha }(G)$. Since each member of the upper central series is a functionally defined subgroup, it is characteristic. In fact, from the way we have defined the members, each member is also strictly characteristic.

Related group properties

• Nilpotent group: If there is a finite $c$ for which $G=Z^{c}(G)$, then $G$ is termed nilpotent of nilpotence class $c$.

• Hypercentral group: In general, the limit of all the $Z^{\alpha }(G)$ is termed the hypercenter of $G$. If $G$ equals its hypercenter.

Relation with upper central series

For a nilpotent group, the lower central series and upper central series are closely related.