H is a subgroup of G iff H is a sunset of G and H satisfies all the axioms of a group using the multiplication rule of G.

Theorems about subgroups:

1. If H and K are subgroups of G then H intersect K is a subgroup of G.
2. If H is a subgroup of G and N is a normal subgroup of G, then H
3. intersect N is a normal subgroup of H.
4. If H is a subgroup of G and N is a normal subgroup of G, then NH=HN
5. is a subgroup of G, not necesserily normal.
6. If H and N are both normal subgroups of G, then H intersect N, and
7. HN are both normal subgroups of G.