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66. 67. (is. 69. 70. 71. 72. 73. 74. Find a noncyclic subgroup of order 4 in U (40). w={[::] b {[3 dlEG | a+b+e+d = 0}. ProvethatHisasubgroupofG. C What if 0 is replaced by 1? Let H = {A E GL(2, R) | detA is an integer power of 2}. Show that H is a subgroup of GL(2, R). Let H be a subgroup of R under addition. Let K = {2\" | a E H}. Prove that K is a subgroup of R* under multiplication. a, b, c, d E Z} under addition. Let H : Let G be a group of functions from R to R*, where the operation of G is multiplication of functions. Let H : {f E G | f(2) : 1}. Prove that H is a subgroup of G. Can 2 be replaced by any real number? 0 Let G : GL(2, R) and H = {[3 bl a andb are nonzero inte gers} under the operation of matrix multiplication. Prove or disprove that H is a subgroup of GL(2, R). LetH : {a + hi I a, b E R, ab 3: 0}. Prove or disprove thatHis a subgroup of C under addition. LetH : {a + bi | a. b E R, a2 + b2 : 1}. Prove or disprove that H is a subgroup of {3* under multiplication. Describe the elements of H geometrically. Let G be a nite Abelian group and let a and b belong to G. Prove