Group Direct Product is Product in Category of Groups

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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}\xymatrix@L+3mu@R=3em{

& 

F
 \ar@/_/[dl]_*{g}
 \ar[d]^*{\quad g \times h}
 \ar@/^/[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

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