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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.

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