Abstract Algebra Theory and Applications - Computer Science ...

EXERCISES 165has kernel H. Hence, H must be normal in G.□Notice that in the course of the proof of Theorem 9.11, we have alsoproved the following theorem.Theorem 9.12 (Third Isomorphism Theorem) Let G be a group andN and H be normal subgroups of G with N ⊂ H. ThenG/H ∼ = G/NH/N .Example 14. By the Third Isomorphism Theorem,Z/mZ ∼ = (Z/mnZ)/(mZ/mnZ).Since |Z/mnZ| = mn and |Z/mZ| = m, we have |mZ/mnZ| = n.Exercises1. For each of the following groups G, determine whether H is a normal subgroupof G. If H is a normal subgroup, write out a Cayley table for thefactor group G/H.(a) G = S 4 and H = A 4(b) G = A 5 and H = {(1), (123), (132)}(c) G = S 4 and H = D 4(d) G = Q 8 and H = {1, −1, i, −i}(e) G = Z and H = 5Z2. Find all the subgroups of D 4 . Which subgroups are normal? What are allthe factor groups of D 4 up to isomorphism?3. Find all the subgroups of the quaternion group, Q 8 . Which subgroups arenormal? What are all the factor groups of Q 4 up to isomorphism?4. Prove that det(AB) = det(A) det(B) for A, B ∈ GL 2 (R). This shows thatthe determinant is a homomorphism from GL 2 (R) to R ∗ .5. Which of the following maps are homomorphisms? If the map is a homomorphism,what is the kernel?(a) φ : R ∗ → GL 2 (R) defined byφ(a) =( 1 00 a)