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,
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 ?
I am not sure if I am showing enough in the proof I wrote down. Some help would be great.