Group Direct Product is Product in Category of Groups

Theorem

Let $\mathbf{Grp}$ be the category of groups.

Let $G$ and $H$ be groups, and let $G \times H$ be their direct product.

Then $G \times H$ is a binary product of $G$ and $H$ in $\mathbf{Grp}$.

Proof

Let $F$ be a group.

By Direct Product of Group Homomorphisms is Homomorphism, given group homomorphisms:

$g: F \to G, h: F \to H$

their direct product $g \times h: F \to G \times H$ is a group homomorphism.

From Projections on Direct Product of Group Homomorphisms, the following diagram is commutative:

$\begin{xy}\[email protected][email protected]=3em{ & F \[email protected]/_/[dl]_*{g} \ar[d]^*{\quad g \times h} \[email protected]/^/[dr]^*{h} \\ G & G \times H \ar[l]^*{\pr_1} \ar[r]_*{\pr_2} & H }\end{xy}$

By Cartesian Product is Set Product, $g \times h$ is the only mapping $F \to G \times H$ that could fit into the diagram.

Since it is also a group homomorphism, we see that the UMP for the binary product is satisfied.

Thus the direct product $G \times H$ indeed is a categorical product for $G$ and $H$ in $\mathbf{Grp}$.

$\blacksquare$

Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 2.5.2$