Let and be groups, with and normal. Prove that is normal and that .

We have already seen that is a subgroup. Now let . we have , hence is normal.

Define by . Clearly is surjective. We now show that . If , we have , hence and . Thus . If , then . Thus , and by the First Isomorphism Theorem, .

## Comments

Where have we already seen that (C x D) is a subgroup. I know I did. I just don’t remember where. Thanks

I can’t recall which problem this was in, but it’s not so hard to see using the subgroup criterion: contains and thus is not empty, and if , then .