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 a composition in .
Since and and and and are subgroups of we see that and similarly .
Thus, , which establishes that is a subgroup of .
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 belong 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 be two subgroups of a group, then is a subgroup if and only if either or .