Abstract Algebra Theory and Applications - Computer Science ...

52 CHAPTER 2 GROUPS32. Find all the subgroups of Z 3 × Z 3 . Use this information to show that Z 3 × Z 3is not the same group as Z 9 .33. Find all the subgroups of the symmetry group of an equilateral triangle.34. Compute the subgroups of the symmetry group of a square.35. Let H = {2 k : k ∈ Z}. Show that H is a subgroup of Q ∗ .36. Let n = 0, 1, 2, . . . and nZ = {nk : k ∈ Z}. Prove that nZ is a subgroup ofZ. Show that these subgroups are the only subgroups of Z.37. Let T = {z ∈ C ∗ : |z| = 1}. Prove that T is a subgroup of C ∗ .38. Let G consist of the 2 × 2 matrices of the form( )cos θ − sin θsin θcos θwhere θ ∈ R. Prove that G is a subgroup of SL 2 (R).39. Prove thatG = {a + b √ 2 : a, b ∈ Q and a and b are not both zero}is a subgroup of R ∗ under the group operation of multiplication.40. Let G be the group of 2 × 2 matrices under addition and{( ) }a bH =: a + d = 0 .c dProve that H is a subgroup of G.41. Prove or disprove: SL 2 (Z), the set of 2 × 2 matrices with integer entries anddeterminant one, is a subgroup of SL 2 (R).42. List the subgroups of the quaternion group, Q 8 .43. Prove that the intersection of two subgroups of a group G is also a subgroupof G.44. Prove or disprove: If H and K are subgroups of a group G, then H ∪ K is asubgroup of G.45. Prove or disprove: If H and K are subgroups of a group G, then HK = {hk :h ∈ H and k ∈ K} is a subgroup of G. What if G is abelian?46. Let G be a group and g ∈ G. Show thatZ(G) = {x ∈ G : gx = xg for all g ∈ G}is a subgroup of G. This subgroup is called the center of G.