Q: Prove that if

and

are normal subgroups of a group

, then

.

A: Since

and

,

and

have the same identity element

. Thus,

which shows

is not empty.

Now, let

. Then

and

. So,

.

Mistake here. Read your line above! In any case, what is the conclusion of this calculation? You want to show that .
If we let

, then

. So, HK is a subgroup of H.

Now, to show it is a normal subgroup, do I just compute

and show it is equal to

?

Yes! You can just use the fact that ...
I am not sure if I am showing enough in the proof I wrote down. Some help would be great.

Thanks