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(6) Any subgroup of the group ℚ under addition is, (A) cyclic & finitely generated but not abelian & normal, (B) cyclic & abelian but not finitely generated & normal, (C) abelian & normal but not cyclic & finitely generated., (D) finitely generated & normal but not cyclic & abelian., , (7) Let σ and τ be the permutations defined by, 1, σ=(, 1, , 2 3 4 5 6 7 8 9, ), 3 5 7 9 6 4 8 2, 1 2 3 4 5 6 7 8 9, and τ = (, ), 7 8 3 4 9 6 5 2 1, Then, (A) σ and τ generate the group of permutations on {1,2,3,4,5,6,7,8,9}, (B) σ is contained in the group generated by τ, (C) τ is contained in the group generated by σ, (D) σ and τ are in the same conjugacy class, (8) Let 𝐺 be a group such that 𝑎2 = 𝑒 for each 𝑎 ∈ 𝐺, where 𝑒 is the identity, element of 𝐺. Then, (A) 𝐺 is cyclic, , (B) 𝐺 is finite, , (C) 𝐺 is abelian, , (D) 𝐺 has a subgroup which is not normal., , 𝑎 𝑏, (9) Let 𝑆 = {(, ) : 𝑎, 𝑏, 𝑐 ∈ ℝ} be the ring under matrix addition and, 0 𝑐, 0 𝑝, multiplication. Then the subset {(, ) : 𝑝 ∈ ℝ} is, 0 0, (A) not an ideal of 𝑆, (B) an ideal but not a prime ideal of 𝑆, (C) is a prime ideal but not a maximal ideal of 𝑆, (D) is a maximal ideal of 𝑆

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(10) Let PID, ED, UFD denote the set of all principal ideal domains, Euclidean, domains, unique factorization domains, respectively. Then, (A) UFD ⊂ ED ⊂ PID, , (B) PID ⊂ ED ⊂ UFD, , (C) ED ⊂ PID ⊂ UFD, , (D) PID ⊂ UFD ⊂ ED, , (11) If 𝑝 is prime, and ℤ𝑝4 denote the ring of integers modulo 𝑝4 , then the, number of maximal ideals in ℤ𝑝4 is, (A) 4, , (B) 2, , (C) 3, , (D) 1, , (12) Which of the following statements is true about 𝑆 = ℤ[𝑥 ]?, (A) 𝑆 is an Euclidean domain since all its ideals are principal, (B) 𝑆 is an Euclidean domain since ℤ is an Euclidean domain, (C) 𝑆 is not Euclidean domain since 𝑆 is not even an integral domain., (D) 𝑆 is not an Euclidean domain since it has non-principal ideal, , (13) The number of elements of a principal ideal domain (PID) can be, (A) 15, , (B) 20, , (C) 25, , (D) 30, , (C) 13, , (D) 03, , (14) The last two digits of 132041 are, (A) 01, , (B) 10, , (15) Which of the following can be the class equation of a group?, (A) 10=1+1+1+2+5, , (B) 4=1+1+2, , (C) 8=1+1+3+3, , (D) 6=1+2+3