Let G be a group and H, K≤ G normal subgroups of G. Prove HnK≤G.

Prove that If |G|=np with p>n and p prime, and H is a subgroup of G with order p, then: H is normal in G.   Please be as clear as possible showing and explaining all the steps, and use definitions if necessary. Thank you very much.

The intersection of any two normal subgroup of a group G is a normal subgroup of G.

Let G be a group, let H≤G be a subgroup, and let NG be a normal subgroup. (i) Show that HnN is a normal subgroup of H. (ii) Suppose now that H is a normal subgroup of G. Show that HnN is a normal subgroup of G.