MCQS

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 

A
 

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.

C. 2 

D. ∞ 

See also  Computer Data Base Management Mcqs
 
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