Direct Product of Central Subgroups

Theorem

Let $G$ and $H$ be groups.

Let $Z$ and $W$ be central subgroups of $G$ and $H$ respectively.

Then $Z \times W$ is a central subgroup of $G \times H$.

Proof

Let $\tuple {z, w} \in Z \times W$.

Let $\tuple {x, y} \in G \times H$.

Then:

\(\ds \tuple {x, y} \tuple {z, w}\)

\(=\)

\(\ds \tuple {x z, y w}\)

Definition of Group Direct Product

\(\ds \)

\(=\)

\(\ds \tuple {z x, w y}\)

Definition of Central Subgroup

\(\ds \)

\(=\)

\(\ds \tuple {z, w} \tuple {x, y}\)

Definition of Group Direct Product

$\tuple {x, y}$ is arbitrary.

Thus $\tuple {z, w}$ commutes with all elements of $G \times H$.

Hence $\tuple {z, w}$ is in the center of $G \times H$

That is, $Z \times W$ is a subgroup of the center $G \times H$.

The result follows by definition of central subgroup.

$\blacksquare$