51. ”Both the order and index of a subgroup of a finite group divides the order of the group” is the statement of:
A. Division Algorithm
B. Lagrange Theorem
C. Euclid Theorem
D. Cayley Theorem
52. The order of an element of a finite group divides:
A. the order of group
B. the order of subgroup
C. the index of every subgroup
D. None of these
53. A group of order … is always cyclic.
A. 7
B. 8
C. 9
D. 10
54. A finite group of … order is necessarily cyclic.
A. Prime
B. Even
C. Odd
D. Composite
55. Which of the following abelian group is not cyclic?
A. (Z, +)
B. (Q, +)
C. (R, +)
D. Both B and C
56. Let G be a group of order 90. G can have a subgroup of order:
A. 30
B. 40
C. 50
D. 60
57. Let G be a cyclic group of order n generated by a. Then for any 1 ≤ k < n, the order of a k is:
A. k/gcd(n,k)
B. n/lcm(n,k)
C. n/gcd(n,k)
D. k/lcm(n,k)
58. Let G be a cyclic group of order 24 generated by a. Then the order of a 10 is:
A. 6
B. 12
C. 18
D. 24
59. Let H and K be two finite subgroups of a group G whose orders are relatively prime, then H ∩ K equals:
A. {e, a}
B. H ∪ K
C. HK
D. {e}
60. Let X be a nonempty set. A bijective function f : X → X is called a … on X.
A. Homomorphism
B. Isomorphism
C. Endomorphism
D. Permutation
