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Abstract Algebra — Exercise Sheet 3 Homomorphisms 1. Let φ : (G, ·) → (L, ·) be a group homomorphism. Prove that φ(eG ) = eL and φ(g −1 ) = φ(g)−1 for all g ∈ G. 2. a) Check that f : (R, +) → (R∗ , ·) given by f (x) = 10x is an injective group homomorphism. Is it surjective? b) Is log : ((0, ∞), ·) → (R, +) a group homomorphism? Is it an isomorphism? 3. Let (G, ·) be a group and consider a new operation ⊙ on G given by a ⊙ b = b · a. Prove that the inverse function defines an isomorphism of groups (G, ·) → (G, ⊙) a → a′ . 4. Given any constant c ∈ R, prove that the multiplication map c : (Rn , +) → (Rn , +) (x1 , x2 , ..., xn ) → (cx1 , cx2 , ..., cxn ) is a morphism of groups. 5. For each of the following functions, check whether it is a homomorphism: a) f : (Z, +) → (Z, +) given by f (n) = n3 . b) g : (Z, +) → (Z3 , +) given by g(n) = n3 ( mod 3). c) h : (Z, +) → (Zp , +) given by h(n) = np ( mod p). Here p is a prime number. 6. Cayley’s theorem. Let (G, ⋆) be a finite group with n elements, G = {g1 , g2 , ..., gn }. Prove that there exists an injective group homomorphism (G, ·) → (Sn , ◦). [Hint: We have shown in class that for each element a ∈ G, the function a : G → G given by b → a ⋆ b is bijective.] 7. Consider the map f : Z3 → Z3 given by f (n) = n + 1. Define a new operation ∗ on Z3 such that f : (Z3 , ∗) → (Z3 , ·) is a monoid isomorphism. 8. Compare the Cayley tables for the group of symmetries of a rhombus, the group of symmetries of a rectangle, and (Z2 × Z2 , +), where + denotes the piecewise addition (a1 , a2 ) + (b1 , b2 ) = (a1 + b1 , a2 + b2 ). Are these groups isomorphic? 9. The symmetry group D4 of a square has eight elements. The quaternion group H = {1; i; j; k; −1; −i; −j; −k} (with 1 being the identity element, i2 = j 2 = k2 = −1 and ij = k; jk = i; ki = j; ji = −k; kj = −i, and ik = −j), also has eight elements. Are they isomorphic? 10. Let f : (G, ∗) → (L, ⋆) be a group homomorphism and define the kernel of the homomorphism to be Kerf := f −1 (eL ) = {x ∈ G; f (x) = eL }. a) Prove that Kerf is a subgroup of G. b) Prove that f is injective ⇐⇒ Kerf = {eG }. 11. Let f : (G, ∗) → (L, ⋆) be a group homomorphism and define the image Imf := f (G) = {f (x); x ∈ G}. Prove that Imf is a subgroup of L. 12. Let (G, ·) be a non-commutative group and let Aut(G) = {f : G → G; f group isomorphism }. For any g ∈ G, consider the function ϕg : (G, ·) → (G, ·) x → gxg−1 . This map is called conjugation by g. a) Prove that (Aut(G), ◦) is a group. It is called the group of automorphisms of G. What is Aut(Z, +)? What is Aut(Z4 , +)? b) Prove that ϕg ∈ Aut(G) for all g ∈ G. c) Prove that there is a homomorphism ϕ : G → Aut(G) given by ϕ(g) = ϕg for all g ∈ G. d) Prove that Kerϕ = {g ∈ G ; gx = xg∀x ∈ G}. This is called the centralizer of G. The centralizer of an abelian group is the entire group. 13. Let (GL2 (R), ·) be the group of 2 × 2 matrices with non-zero determinant (see Q13, Ex. Set 4). Prove that the function f : (C∗ , ·) → (GL2 (R), ·) given by x y x + iy → −y x is an injective group homomorphism. 14. Prove that any cyclic group is isomorphic to either Z or Zn for some n ∈ N. As a corollary, prove that all the subgroups of a cyclic group are cyclic. 15. Let X = {(12)(34), (13)(24), (23)(14)} denote the set of three pairs of opposite sides in a tetrahedron (triangular pyramid). Each element σ ∈ S4 defines a permutation σ:X → X (ij)(kl) → (σ(i)σ(j))(σ(k)σ(l)). Use this to construct a map F : S4 → S3 . Find KerF . 1 b 4 b 2 b 3 b Note: Alternatively, you can consider X = {(12)(34), (13)(24), (23)(14)} ⊂ S4 . For every σ ∈ S4 , prove that the map ϕσ : S4 → S4 given by ϕσ (x) = σxσ −1 satisfies ϕσ (X) = X. Hence, ϕσ defines a bijective map from X to X. Check that this is the same as the map F : S4 → S3 . (See Q11 in Ex.Set3). 16. Consider the operations ⊞ and ⊠ : Z6 × Z6 → Z6 given by a ⊞ b := a + b + 1, a ⊠ b := 5ab + 5(a + b) + 4. Prove that (Z6 , ⊞) and (Z6 , ⊠) are monoids. What is the identity element in each case? Is ⊠ distributive with respect to ⊞? Let + and × be the usual addition and multiplication on Z6 . Find c ∈ Z6 such that f (n) = 5n+c defines an isomorphism from (Z6 , +) to (Z6 , ⊞) and from (Z6 , ×) to (Z6 , ⊠). 17. Let (Mn×n (R), ·) denote the set of n × n matrices with real entries. Given A ∈ Mn×n (R), we denote by Aij the entry on the i-th row and j-th column of A. We define the product AB by (AB)ij = n X Aik Bkj = Ai1 B1j + Ai2 B2j + · · · Ain B2n . k=1 For any matrix A ∈ Mn×n (R), let At denote the transpose matrix, obtained from A by swapping rows with columns Prove that the transpose function defines an isomorphism of monoids (Mn×n (R), ·) → (Mn×n (R), ⊙) A → At . where by definition A ⊙ B = BA. [Hint: See solutions of Q14 in Ex. Set 4] 18. Prove that the determinant function A → detA is a group homomorphism det : (GL2 (R), ·) → (R∗ , ·). 19. a) Prove that f : (Sn , ◦) → (GLn (R), ·) given by α → f (α), where f (α)ij = 1 if i = α(j), 0 otherwise , is an injective homomorphism. b) If n = 3 and α = (132), check that det(f (α)) = sign (α). 20. Consider the function f : Z → Z13 × Z11 given by f (a) = (a( mod 13), a( mod 11)). • Check that f : (Z, +) → (Z13 × Z11 , +) is a group homomorphism. • Prove that Ker(f ) = 143Z. • Prove that ([1], [0]) and ([0], [1]) are in the image of f . Hence or otherwise, prove that f is surjective and (Z143 , +) ∼ = (Z13 × Z11 , +). • Generalize the result above by replacing 13 and 11 with any two relatively prime numbers. Prove your results.