Quotient Group of Direct Products

Theorem
Let $G$ and $G'$ be groups.

Let:
 * $H \lhd G$
 * $H' \lhd G'$

where $\lhd$ denotes the relation of being a normal subgroup.

Then:
 * $(1): \quad \left({H \times H'}\right) \lhd \left({G \times G'}\right)$
 * $(2): \quad \left({G \times G'}\right) / \left({H \times H'}\right)$ is isomorphic to $\left({G / H}\right) \times \left({G' / H'}\right)$

where:
 * $H \times H'$ denotes the group direct product of $H$ and $H'$
 * $G / H$ denotes the quotient group of $G$ by $H$.

Proof
$(1): \quad \left({H \times H'}\right) \lhd \left({G \times G'}\right)$:

Let $\left({x, x'}\right) \in G \times G'$ and $\left({y, y'}\right) \in H \times H'$.

Then:

so $\left({H \times H'}\right) \lhd \left({G \times G'}\right)$.

$(2): \quad \left({G \times G'}\right) / \left({H \times H'}\right)$ is isomorphic to $\left({G / H}\right) \times \left({G' / H'}\right)$:

Let $\varphi_1 : G \to G / H$ and $\varphi_2 : G' \to G' / H'$ be the quotient epimorphisms with $H$ and $H'$ as their kernels, respectively.

Now define a homomorphism $\varphi : G \times G' \to \left({G / H}\right) \times \left({G' / H'}\right)$ by $\varphi = \varphi_1 \times \varphi_2$, so $\varphi((x,x')) = (\varphi_1(x), \varphi_2(x'))$.

The kernel of $\varphi$ is clearly $H \times H'$, and $\varphi$ is surjective.

So $\left({G / H}\right) \times \left({G' / H'}\right) \cong \left({G \times G'}\right) / \left({H \times H'}\right)$ by the first isomorphism theorem for groups.