A group, G, is a set with a multiplication rule defined on it which must satsfy:

1. gi*gj in G forall gi*gj (closure)
2. gi*(gj*gk)=(gi*gj)*gk (associativity)
3. There exists identity, e, s.t. e*g=g=g*e
4. forall g in G there exists g-1 (g inverse) s.t. g*g-1=g-1*g=e

Groups express symmetries. There are many examples of symmetries in nature and this is why groups are useful in physics.

Some theorems:

• H,K subgroups of G => H intersect K subgroup of G
• H,N subgroups of G with N normal => H intersect N subgroup of H is normal w.r.t H
• H,N normal subgroups of G => H intersect N subgroup of H is normal w.r.t G