Alg7.13

From Exampleproblems

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.