For (a), it would be helpful to use the following lemma.

Lemma. If M and K are subgroups of a group H, MK is a subgroup of H if and only if MK = KM.

Now you need to show that and in order to apply the above lemma. I'll show and I'll leave it to you to show that the reverse inclusion.

Let such that . Since M is a normal subgroup of H, . Thus .

(b) You might need to use this.