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:
 * $\paren {G \times G'} / \paren {H \times H'}$ is isomorphic to $\paren {G / H} \times \paren {G' / H'}$

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

Proof
Let $\phi_1: G \to G / H$ and $\phi_2: G' \to G' / H'$ be the quotient epimorphisms with $H$ and $H'$ as their kernels, respectively.

Now define a homomorphism $\phi: G \times G' \to \paren {G / H} \times \paren {G' / H'}$ by:
 * $\phi = \phi_1 \times \phi_2$

so:
 * $\map \phi {\tuple {x, x'} } = \tuple {\map {\phi_1} x, \map {\phi_1} {x'} }$

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

So by the First Isomorphism Theorem for Groups:
 * $\paren {G / H} \times \paren {G' / H'} \cong \paren {G \times G'} / \paren {H \times H'}$