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Modern Algebra K1 Level Questions UNIT I

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Modern Algebra K1 Level Questions UNIT I Modern algebra K1 Level Questions UNIT I 1. If order of group G is ,where p is prime then a. G is abelian b b. G is not abelian c. G is ring d. None of these 2. If G is a group, for a G ,N(a) is the normalize of a then x N(a) a. = ∈ ∀ ∈ b. = c. = . d. ̟ p 3. If G is̟ a groupp , then for all a, b G ̟≠ ̟ a. ( ) = b. ( ) = ̟ ̟ c. = . . d. = ̟ ̟ 4. If G is a set of integers and . , then G is ̟ ̟ a. Quasi-group ̟ ≡ ̟ − b. Semi group c. Monoid d. Group 5. In a group G , for each element a G , there is a. No inverse ∈ b. A unique inverse c. More than one inverse . ∈ d. None of these 6. If p is a prime number and | ( ) , a. b f ℎpc ̟ ∈ b. G . ∈ c. ̟ ∉ d. ̟ ⊂ 7. If G is a group of order then , order of identity element is ̟ ⊃ a. One c b. Greater than one c. n d. none of these 8. If is of order is prime to n , then order of is a. n ̟ ∈ c ̟cȍ b ̟ b. one c. less than n d. greater than n 9. If the order of elements , are respectively then a. = ̟ ̟ ∈ ̟cȍ c b. c. Ţ = = 0 d. ≠ c 10. If in a groupc , ,the order of a is n and order of is m then cfcp fŲ ℎpıp a. ̟ ∈ ̟ b. c. Ţ =≤ d. ≥ c 11. The identityc permutation is cfcp fŲ ℎpıp a. Even permutation b. Odd permutation c. Neither even nor odd d. None of these 12. The product of even permutation is a. Even permutation b. Odd permutation c. Neither even nor odd d. None of these 13. The inverse if an even permutation is a. Even permutation b. Odd permutation c. Even or odd permutation d. None of these 14. The product of (1 2 4 5)(3 2 1 5 4) a. ( ) aı b. (1 5) c. (3 4 1) d. (1 5 3 1) 15. The inverse of an odd permutation is a. Even permutation b. Odd permutation c. Even or odd permutation d. None of these 16. If are the inverse of some elements , then a. = ̟cȍ ̟ ∈ Ʀ b. c. = , d. ≠ ŲfDz ı̟p cfcp fŲ ℎpıp 17. Let Z be a set of integers , then under ordinary multiplication(Z,·) is a. Monoid b. Semi group c. Quasi group d. Group 18. is a sub group of G if a. = b. c. d. ⊂ ⊃ 19. If G = 1, 1, is a multiplicative group then order of – is ≠ a. One − a − a a b. Two c. Three d. Four 20. If G is a group of even order , a≠ e if = then G is a. Abelian group ∀ ̟ p b. Sub group c. Normal group d. None of these UNIT II 1. If G = 0 ,1,2,3,4 , + the order of 2 is a. One b. Two c. Three d. Four 2. Every group of prime order is a. Cyclic b. Abelian c. Sub group d. Normal group 3. The number of elements in a group is a. Identity of group b. Order of group c. Inverse of group d. None of these 4. A – mapping of finite group onto itself is a. Isomorphism fcp fcp b. Homomorphism c. Automorphism d. None of these 5. If in a group , a. ( ) = ∀ ̟ ∈ b. ( )= ̟ ̟ c. ( )= ̟ ̟ d. 6. If =̟ 2 3 and̟ = (4 5) be two permutations on five symbols 1,2,3,4,5 is Ưfcp fŲ ℎpıp a. Ų ℎpc Ų b. c. d. 7. Given permutation is equivalent to a. (1 6 3 2)(2 1) b. (1 6 3 2)(1 1) c. (1 6 3 2)(4 5) d. (1 6 3 2)(5 4) 8. If the given permutation are = , = find a. b. c. d. 9. If number of left cosets of H in G are and the number of right cosets of H in G are then a. = c b. c. Ţ ≥ c ≤ c d. 10. If H is a sub group of finite group G and order of H and G are respectively then Ưfcp fŲ ℎpıp a. | ̟cȍ c b. | c. Ţ d. c ∤ c Ưfcp fŲ ℎpıp 11. If is a finite group of order ,then for every a G , we have a. = an identity element c ∈ b. = . c. = ̟ ̟ d. 12. If ̟ ̟ are two sub groups of G then following is also sub group of G Ưfcp fŲ ℎpıp a. ̟cȍ b. c. ∩ d. ∪ 13. The set M of square matrices (of same order) with respect to matrix multiplication is Ưfcp fŲ ℎpıp a. Group b. Semi group c. Monoid d. Quasi group 14. If (G, ) is a group and , G = , then is a. Abelian group ∗ ∀ ̟ ∈ ∗ ̟ ∗ ∗ ̟ p b. Non abelian group c. Ring d. Field 15. If G is a group such that = , ,then G is a. Abelian group ̟ p ∀ ̟ ∈ b. Non abelian group c. Ring d. Field 16. If = (2 3) and = (4 5) are two permutations on 1 , 2, 3, 4, 5 then is a. Ų Ų b. c. d. 17. If is the order of element of group G then = , an identity element if a. | c ̟ ̟ p b. | c. c Ţ d. c c ∤ 18. The order of element in a group G is a. One b. Zero c. Order of a group d. Less than order of a group 19. If , a group and order of are respectively then a. > ̟ ̟ ∈ ̟ ̟cȍ ̟ ̟cȍ c b. < c. = c d. c 20. If ,Ţ , a group of order m then order of are Ưfcp fŲ ℎpıp a. Same ̟ ∈ ̟̟cȍ ̟ b. Equal to m c. Unequal d. None of these UNIT III 1. If G = 1, 1 is a group , then the order of 1 is a. One − b. Two c. Zero d. None of these 2. The product of permutations (1 2 3 ) (2 4 3) (1 3 4)is equal to a. ∙ ∙ b. c. d. 3. The permutations is equal to a. (1 5)(1 3)(2 4) b. (1 )(2)(3) c. (1 3 5)(5 6) d. (1 4 2)(5 3) 4. Given the permutation = (1 2 3 4 5 6 7) then is a. (1 3 5 7 2 4) b. (1 4 7 3 6 2 5 ) c. (1 7 6 5 4 3 2 ) d. 5. If = 1 2 3 4 is a. (1 3 )(2 4) ℎpc b. (1 3) c. (2 4) d. (2 3)(3 1) 6. Statement A : All cyclic group are abelian Statement B: The order of cyclic group is same as the order of its generator a. A and B are False b. A is true and B is false c. B is true , A is false d. A and B are true 7. Statement A : Every isomorphic image of a cyclic group is cyclic Statement B: Every homomorphic image of a cyclic group is cyclic a. Both A and Bare true b. Both A and B are false c. A is true only d. B is true only 8. A element of an finite cyclic group G of order n is a generator of if 0< < and also a. is prime to ̟ b c b. is the multiple of c. is the multiple of b c d. None of these c c 9. If is the finite group of order n, and order of a is , if is the cyclic then a. > ̟ ∈ b. < c. Ţ = c d. c 10. If isc a generator of a cyclic group and order of a is < then order of a cyclic group Ưfcp fŲ ℎpıp m is ̟ ∈ c ∞ a. b. = c. cŲaca> Ţ c d. < 11. Let G = 1, 1 then under ordinary multiplication (G , ) is c a. Monoid − ∙ b. Semi group c. Quasi group d. Group 12. Let be a set of rational numbers under ordinary addition ( , + ) is a. Monoid b. Semi group c. Quasi group d. Group 13. Let be a group of square matrices of same order with respect to matrix multiplication then It is not a a. Quasi group b. Abelian group c. Semi group d. None of these 14. If is a finite group , then for every then order is a. Finite ̟ ∈ ̟ b. Infinite c. Zero d. None of these 15. In the additive group of integers , the order of every element 0 is a. Infinity ̟ ≠ b. One c. Zero d. None of these 16. In a additive group of integers , the order of identity element is a. Zero b. One c. Infinity d. None of these 17. In the additive group G of integers the order of inverse element , is a. Zero ̟ ∀ ̟ ∈ b. One c. Infinity d. None of these 18. The singleton 0 with binary operations addition and multiplication is ring and it is called a. Zero ring b. Division ring c. Singleton ring d. None of these 19. The element 0 , the commutative ring is an integral domain if a. = 0, = 0 ̟ ≠ ∈ b. = , c. ̟ , ∈ ̟cȍ = 0 . ∈ .Ƒ≠ d. = 0 , = ̟≠ f ∈ ̟cȍ ̟ ∈ ̟cȍ f 20. A ring is a integral domain if a. is a commutative ring b. is a commutative ring with zero divisor c. is a commutative ring with non zero divisor d. is a ring with zero divisor UNIT IV 1.
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