Abstract Algebra Theory and Applications - Computer Science ...

166 CHAPTER 9 HOMOMORPHISMS AND FACTOR GROUPS(b) φ : R → GL 2 (R) defined byφ(a) =(1 0a 1)(c) φ : GL 2 (R) → R defined by(( a bφc d(d) φ : GL 2 (R) → R ∗ defined by(( a bφc d))= a + d))= ad − bc(e) φ : M 2 (R) → R defined by(( a bφc d))= b,where M 2 (R) is the additive group of 2 × 2 matrices with entries in R.6. Let T be the group of nonsingular upper triangular 2×2 matrices with entriesin R; that is, matrices of the form( ) a b,0 cwhere a, b, c ∈ R and ac ≠ 0. Let U consist of matrices of the form( ) 1 x,0 1where x ∈ R.(a) Show that U is a subgroup of T .(b) Prove that U is abelian.(c) Prove that U is normal in T .(d) Show that T/U is abelian.(e) Is T normal in GL 2 (R)?7. Let A be an m×n matrix. Show that matrix multiplication, x ↦→ Ax, definesa homomorphism φ : R n → R m .8. Let φ : Z → Z be given by φ(n) = 7n. Prove that φ is a group homomorphism.Find the kernel and the image of φ.9. Describe all of the homomorphisms from Z 24 to Z 18 .