The problem 1 is proved by the definition of group (satisfying closure, association law, existence of unit element and existence of inverse element).

The second question is also proved by the definition of group. Because other properties are obvious, it is only necessary to prove the existence of unit elements:

A∈H, b∈H, according to the closure,

a∈H

Then e=aa∈H

That is, the unit element e of g must also be in H.