Group Epimorphism preserves Central Subgroups

Theorem

Let $G$ and $H$ be groups.

Let $\theta: G \to H$ be an epimorphism.

Let $Z \le G$ be a central subgroup of $G$.

Then $\theta \sqbrk Z$ is a central subgroup of $H$.

Proof

By definition of central subgroup:

$Z \subseteq \map Z G$

where $\map Z G$ denotes the center of $G$.

From Image under Epimorphism of Center is Subset of Center:

$\theta \sqbrk {\map Z G} \subseteq \map Z H$

From Image of Subset under Mapping is Subset of Image it follows that:

$\theta \sqbrk Z \subseteq \map Z H$

The result follows.

$\blacksquare$