# Alg7.13

If $G_{1}$ is a subgroup of $G$ and $H_{1}$ is a subgroup of $H$, prove $G1\times H1$ is a subgroup of $G\times H$.
We know $G_{1}$, $H_{1}$, $G$, and $H$ are all groups. Thus, since the Cartesian product of any two groups is a group, $G_{1}\times H_{1}$ and $G\times H$ are both groups.
A subgroup is merely a subset of elements from a group which, under the same operation as the original group, form a group. $G_{1}$ uses the same operation as $G$ and $H_{1}$ uses the same operation as $H$. Therefore, $G_{1}\times H_{1}$ uses the same operation as $G\times H$. All that is left to be shown is that $G_{1}\times H_{1}$ is a subset of $G\times H$.
Pick any element $(g_{1},h_{1})\in G_{1}\times H_{1}$. Well, $g_{1}\in G$ and $h_{1}\in H$ so $(g_{1},h_{1})\in G\times H$. Thus, every element in $G_{1}\times H_{1}$ is also an element of $G\times H$ and our proof is complete.