# Alg7.5

From Example Problems

Prove that if is a group and are subgroups of , then is a group.

I prove it with the subgroup criteria:

If and , then is a subgroup.

First I prove that which implies .

and . Since . Similarly for . Since .

Second, I prove that .

If then and . This implies and since and are both subgroups of , so . Therefore, is a group.