Group Theory Mcqs With Solution
41. If H and K are subgroups of a group G, then which of the following need not to be a subgroup of G?
A. H ∪ K
B. H ∩ K
C. He
D. eK
42. Let G be a group and G =< a >, for some a ∈ G, then a is called … of G.
A. Involutory
B. Idempotent
C. Generator
D. None of these
43. Let G be a finite group of order n generated by a ∈ G. Then a^i = a^j if and only if:
A. n|(i − j)
B. n|(i + j)
C. i = j
D. None of these
44. Let G be an infinite group generated by a ∈ G. Then a i = a j if and only if:
A. n|(i − j)
B. n|(i + j)
C. i = j
D. None of these
45. Let G be a cyclic group of order 18. How many subgroups of G are of order 6?
A. 1
B. 2
C. 3
D. None of these
46. A partition of a set A is the collection of subsets {Ai : i ∈ I} of A such that
A. A = ∪{Ai : i ∈ I} and Ai ∩ Aj = φ, where i, j ∈ I and i 6= j.
B. A = ∪{Ai : i ∈ I} and Ai ∩ Aj = φ, where i, j ∈ I and i = j.
C. A = ∪{Ai : i ∈ I} and Ai ∩ Aj 6= φ, where i, j ∈ I and i 6= j.
D. A = ∩{Ai : i ∈ I} and Ai ∩ Aj = φ, where i, j ∈ I and i 6= j.
47. Let H be a subgroup of G. Then the set of all left cosets of H in G defines a …on G.
A. Equivalence relation
B. Partition
C. Transitive relation
D. All of these
48. The number of distinct left cosets of a subgroup H of a group G is called the … of H in G, and it is denoted by [G : H].
A. Index
B. Cardinality
C. Order
D. Partition
49. The index of {0̅, 2̅, 4̅} in {0̅, 1̅, 2̅, 3̅, 4̅, 5̅} is:
A. 1
B. 2
C. 3
D. 4
50. The index of {0, ±2, ±4, …} in the group (Z, +) is:
A. 0
B. 1
C. 2
D. ∞