Groups, Subgroups and Cyclic Groups - Question Bank

1. Which of the following is always a subgroup of any group G?
A) The set of all elements of G.
B) The set containing only the identity element.
C) The set of all elements with finite order.
D) The set of all generators.
2. What is the set of all elements 'g' in a group G such that g*x = x*g for all x in G called?
A) The center of G
B) A subgroup of G
C) A cyclic subgroup
D) The identity element
3. If G is a cyclic group and g is a generator, which element is the identity element?
A) g
B) g^0
C) g^(-1)
D) There is no identity element in a cyclic group.
4. What is the order of the element 5 in the group of integers modulo 10 under addition?
A) 1
B) 2
C) 5
D) 10
5. Which of the following sets forms a group under multiplication?
A) The set of all integers
B) The set of all even integers
C) The set of non-zero rational numbers
D) The set {0, 1}
6. Let H be a subgroup of G. If H is cyclic, is G necessarily cyclic?
A) Yes
B) No
C) Only if H=G
D) Only if G is finite
7. What is the smallest group that contains all cyclic groups of order n?
A) The cyclic group of order n.
B) The cyclic group of order 2n.
C) The cyclic group of infinite order.
D) The trivial group.
8. If G is a finite group and 'a' is an element of G, which of the following is always true?
A) The order of 'a' divides the order of G.
B) The order of 'a' is equal to the order of G.
C) The order of 'a' is 1.
D) The order of 'a' is infinite.
9. Consider the group of integers modulo 7 under addition (Z7, +). What is the order of the element 4?
A) 1
B) 3
C) 7
D) 4
10. Which property is NOT required for a set with a binary operation to be a group?
A) Closure
B) Associativity
C) Commutativity
D) Existence of Identity Element
11. In a cyclic group G generated by 'g', what is the order of the element g^k where n = order(g)?
A) n
B) k
C) n / gcd(n, k)
D) gcd(n, k)
12. What is the index of a subgroup H in a group G, denoted by [G:H]?
A) The number of distinct left (or right) cosets of H in G.
B) The order of the subgroup H.
C) The order of the group G.
D) The number of elements in the intersection of H and G.
13. Let G be a group and H a subgroup. If 'a' is an element of G, what is the left coset 'aH'?
A) {ah | h in H}
B) {ha | h in H}
C) {a*h | h in H}
D) {a/h | h in H}
14. What is the identity element for the operation in the group of invertible n x n matrices under matrix multiplication?
A) The zero matrix
B) The identity matrix (I_n)
C) The inverse of the matrix
D) The determinant of the matrix
15. Which of the following is an example of a cyclic group under multiplication?
A) The set of all real numbers
B) The set of all rational numbers
C) The set of nth roots of unity
D) The set of all integers
16. What is the order of the element 1 in the group of non-zero real numbers under multiplication?
A) 1
B) -1
C) Infinite
D) 0
17. If G is a cyclic group generated by 'g', and H is a subgroup of G, what can be said about H?
A) H is always G.
B) H is always the trivial subgroup.
C) H is always cyclic.
D) H is always abelian.
18. What is the order of the group of integers modulo n (Zn, +)?
A) 1
B) n
C) 0
D) Infinite
19. Consider the group of symmetries of an equilateral triangle (D3). Is it a cyclic group?
A) Yes, it is generated by a rotation.
B) Yes, it is generated by a reflection.
C) No, it is not cyclic because it is not abelian.
D) No, it is not cyclic because it has order 6 but no element of order 6.
20. Which of the following is a necessary and sufficient condition for a non-empty subset H of a group G to be a subgroup?
A) For all a, b in H, a * b is in H.
B) For all a in H, a^(-1) is in H.
C) For all a, b in H, a * b^(-1) is in H.
D) H contains the identity element.
21. What is the order of the element 2 in the group Z12 under addition?
A) 2
B) 4
C) 6
D) 12
22. Let G be a cyclic group of order n. How many subgroups does G have?
A) Exactly n subgroups.
B) Exactly one subgroup for each divisor of n.
C) Exactly n-1 subgroups.
D) Only trivial subgroups.
23. Which of the following is a property of cyclic groups?
A) Every cyclic group is abelian.
B) Every abelian group is cyclic.
C) Only finite groups can be cyclic.
D) Cyclic groups must have at least two generators.
24. If o(a) = n, what is the order of the cyclic subgroup generated by 'a'?
A) 1
B) n
C) Infinite
D) n^2
25. What is the order of the element 'a' in a group G, denoted by o(a)?
A) The smallest positive integer n such that a^n = e (identity).
B) The number of elements in the group.
C) The number of elements in the cyclic subgroup generated by 'a'.
D) The inverse of 'a'.
26. Consider the group of integers modulo 10 under addition (Z10, +). Which of the following is NOT a subgroup?
A) {0, 5}
B) {0, 2, 4, 6, 8}
C) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
D) {0, 3, 6, 9, 1, 4, 7}
27. If a subgroup H has index [G:H] = 2 in a group G, what can be said about H?
A) H is always a normal subgroup.
B) H is always a cyclic subgroup.
C) H is always a trivial subgroup.
D) H is always equal to G.
28. Let G be a group. For which property is the subset {e} (where e is the identity) always a subgroup?
A) Closure
B) Associativity
C) Identity
D) Inverse
29. Which element generates the cyclic group Z4 under addition?
A) 0
B) 1
C) 2
D) 3
30. What is the order of the cyclic group Z5 under addition?
A) 1
B) 5
C) 0
D) Infinite
31. What is the relationship between subgroups of a cyclic group?
A) All subgroups are cyclic.
B) Only finite subgroups are cyclic.
C) Subgroups can be non-cyclic.
D) Subgroups only exist if the group is finite.
32. Is the group of non-zero rational numbers under multiplication a cyclic group?
A) Yes
B) No
C) Only if it's finite
D) It depends on the specific elements
33. If a group G is cyclic and generated by element 'g', what is the inverse of g^k?
A) g^k
B) g^(-k)
C) g^(order(g) - k)
D) g^(k - order(g))
34. What is the order of the element 3 in the cyclic group Z6 under addition?
A) 1
B) 2
C) 3
D) 6
35. Which of the following is a cyclic group under addition?
A) The group of rational numbers (Q, +)
B) The group of real numbers (R, +)
C) The group of integers modulo n (Zn, +)
D) The Klein four-group
36. What is a cyclic group?
A) A group where every element is its own inverse.
B) A group that can be generated by a single element.
C) A group where the operation is commutative.
D) A group with a finite number of elements.
37. Which theorem states that the order of a subgroup must divide the order of the group?
A) Lagrange's Theorem
B) Cauchy's Theorem
C) Sylow's Theorem
D) Cayley's Theorem
38. What is the order of the subgroup {0, 3, 6, 9} in the group of integers modulo 12 under addition?
A) 3
B) 4
C) 12
D) 6
39. What is the order of a subgroup?
A) The number of elements in the subgroup.
B) The number of elements in the group.
C) The number of generators of the subgroup.
D) The number of cyclic subgroups.
40. Let G = {1, -1, i, -i} be the group of fourth roots of unity under multiplication. Which of the following is a subgroup?
A) {1, -1}
B) {1, i}
C) {1, -i}
D) {1, -1, i}
41. Consider the group of integers under addition (Z, +). Which of the following is a subgroup?
A) The set of even integers (2Z)
B) The set of positive integers
C) The set of prime numbers
D) The set {0, 1}
42. What are the trivial subgroups of any group G?
A) G and the set containing only the identity element.
B) The set containing only the identity element.
C) G itself.
D) The set containing all elements and the set containing no elements.
43. What is a subgroup?
A) Any subset of a group.
B) A subset of a group that is itself a group under the same operation.
C) A subset of a group that contains the identity element.
D) A subset of a group that contains all inverses.
44. Which property of a group states that for any elements a, b, and c in the group, (a * b) * c = a * (b * c)?
A) Closure
B) Identity
C) Inverse
D) Associativity
45. Is the set of positive integers under addition a group? Why or why not?
A) Yes, it satisfies all group axioms.
B) No, it lacks an identity element.
C) No, it lacks inverse elements.
D) No, it is not closed under addition.
46. Which of the following sets forms a group under addition?
A) The set of odd integers
B) The set of positive integers
C) The set of all integers (Z)
D) The set {0, 1}
47. What is the inverse of element 'a' in the group of non-zero rational numbers under multiplication?
A) a
B) -a
C) 1/a
D) a^2
48. What is the identity element in the group of integers under addition?
A) 1
B) -1
C) 0
D) Does not exist
49. Which operation must be closed under a group (G, *)?
A) For all a, b in G, a * b is in G
B) For all a in G, a * a is in G
C) For all a, b in G, a * b = b * a
D) For all a in G, a * e = a, where e is the identity
50. What is the defining property of a group in abstract algebra?
A) Closure, Associativity, Identity, Inverse
B) Closure, Commutativity, Identity, Inverse
C) Closure, Associativity, Commutativity, Inverse
D) Associativity, Identity, Inverse, Distributivity
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