1. ## Proving subgroup statements

Suppose that H and K are subgroups of a group G. Prove the following statements under additional conditions:

(a). First, give an example that HK may not be a subgroup of G.
(b). Prove that if H is a normal subgroup of G, then HK is a subgroup of G.
(c). Prove that if both H and K are normal subgroups of G, then HK is a normal subgroup of G.

2. ## Re: Proving subgroup statements

Originally Posted by kellsbells92
Suppose that H and K are subgroups of a group G. Prove the following statements under additional conditions:
(a). First, give an example that HK may not be a subgroup of G.
(b). Prove that if H is a normal subgroup of G, then HK is a subgroup of G.
(c). Prove that if both H and K are normal subgroups of G, then HK is a normal subgroup of G.
We are here to help you not to do the work for you.
Therefore, we need to see what effort you have made.
I strongly encourage to do part (a) on your own. It can teach you so much.

If $\displaystyle J\subseteq G~\&~\{a,b\}\subset J$ then if you can show that $\displaystyle ab^{-1}\in J$ that is sufficient to show that $\displaystyle J$ is a subgroup.
Now once you have done (b) then (c) falls out.