__Theorem 1__**:** The intersection of two subgroups of a group is a subgroup of .

**Proof:** Let and be any two subgroups of . Then because at least the identity element is common in both and .

Now to prove that is a subgroup of , it is sufficient to show that , , being composition in .

Since and and and and are subgroups of we see that and similarly .

Thus, , which establishes that is a subgroup of .

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__Theorem 2__**:** The union of two subgroups is not necessarily a subgroup.

**Proof: **For example, let be the additive group of integers, and let

Then are subgroups of , but , which is not a group. It is evident that the closure property is not satisfied. For, 2 + 3 =5, which does not belongs to.

The set is certainly a group.

__Theorem 3__**:** The union of two subgroups is a subgroup if and only if one is contained in the other.

Let are two subgroups of a group, then is a subgroup if and only if, either or .