Quotient Group of Direct Products

Theorem

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

Let:

$H \triangleleft G$

$H' \triangleleft G'$

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

Then:

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

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

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

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

Then:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left({x, x'}\right) \left({y, y'}\right) \left({x, x'}\right)^{-1}\)	\(=\)	\(\displaystyle \left({x, x'}\right) \left({y, y'}\right) \left({x^{-1}, x'^{-1} }\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({x y x^{-1}, x' y' x'^{-1} }\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\in\)	\(\displaystyle H \times H'\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by the normality of $H$ and $H'$.

$(2): \quad \left({G \times G'}\right) / \left({H \times H'}\right)$ 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 $\varphi = \varphi_1 \times \varphi_2$.

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

So $\left({G / H}\right) \times \left({G' / H'}\right) \cong \left({G \times G'}\right) / \left({H \times H'}\right)$.

Sources

• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 47 \theta$