Let φ be a homomorphism from a group G to a group and H be a subgroup of G. Then

(1) is a subgroup of .

(2) If H is cyclic, so is φ(H).

(3) If H is Abelian, so is φ(H).

(4) If H is normal in G, then φ(H) is normal in φ(G).

(5) If |Ker φ| = n, then φ is an n-to-1 mapping from G onto φ(G).