Abstract Algebra Theory and Applications - Computer Science ...

164 CHAPTER 9 HOMOMORPHISMS AND FACTOR GROUPSNow define a map φ from H to HN/N by h ↦→ hN. The map φ is onto,since any coset hnN = hN is the image of h in H. We also know that φ isa homomorphism becauseφ(hh ′ ) = hh ′ N = hNh ′ N = φ(h)φ(h ′ ).By the First Isomorphism Theorem, the image of φ is isomorphic to H/ ker φ;that is,HN/N = φ(H) ∼ = H/ ker φ.SinceHN/N = φ(H) ∼ = H/H ∩ N.ker φ = {h ∈ H : h ∈ N} = H ∩ N,□Theorem 9.11 (Correspondence Theorem) Let N be a normal subgroupof a group G. Then H ↦→ H/N is a one-to-one correspondence betweenthe set of subgroups H containing N and the set of subgroups of G/N.Furthermore, the normal subgroups of H correspond to normal subgroupsof G/N.Proof. Let H be a subgroup of G containing N. Since N is normalin H, H/N makes sense. Let aN and bN be elements of H/N. Then(aN)(b −1 N) = ab −1 N ∈ H/N; hence, H/N is a subgroup of G/N.Let S be a subgroup of G/N. This subgroup is a set of cosets of N. IfH = {g ∈ G : gN ∈ S}, then for h 1 , h 2 ∈ H, we have that (h 1 N)(h 2 N) =hh ′ N ∈ S and h −11 N ∈ S. Therefore, H must be a subgroup of G. Clearly,H contains N. Therefore, S = H/N. Consequently, the map H ↦→ H/H isonto.Suppose that H 1 and H 2 are subgroups of G containing N such thatH 1 /N = H 2 /N. If h 1 ∈ H 1 , then h 1 N ∈ H 1 /N. Hence, h 1 N = h 2 N ⊂ H 2for some h 2 in H 2 . However, since N is contained in H 2 , we know thath 1 ∈ H 2 or H 1 ⊂ H 2 . Similarly, H 2 ⊂ H 1 . Since H 1 = H 2 , the mapH ↦→ H/H is one-to-one.Suppose that H is normal in G and N is a subgroup of H. Then itis easy to verify that the map G/N → G/H defined by gN ↦→ gH is ahomomorphism. The kernel of this homomorphism is H/N, which provesthat H/N is normal in G/N.Conversely, suppose that H/N is normal in G/N. The homomorphismgiven byG → G/N → G/NH/N