If (h, •) and (k, •) are subgroups of (g, •), prove that (h n k, •) is a subgroup of (g, •). can the same be said for the union, hu k? prove or give a counterexample.

Let . Since both and are subgroups of , we have



Because both are subgroups of , then contains the identity element . Furthermore, there must be some element and , i.e. , such that .

Now let . By the same reasoning as above it follows that each of these belong in , and since is a group, we have , so is associative under .

So contains the identity, is closed with respect to inverses, and is associative under . Therefore must be a subgroup of .

In the case of union, this is not always the case. Consider the group with subgroups and where is the set of all integer multiples of 2 and is the set of all integer multiples of 3.

It's easy to show that and are indeed subgroups, but this is not the case for . We have



Take the elements and . Addition yields , but , so is not closed under .