Normal Subgroup and Quotient Groups MCQ Quiz - Objective Question with Answer for Normal Subgroup and Quotient Groups - Download Free PDF
Last updated on May 19, 2025
Latest Normal Subgroup and Quotient Groups MCQ Objective Questions
Normal Subgroup and Quotient Groups Question 1:
Let G be a group and N a subgroup of G . Which of the following statements is true about N being a normal subgroup of G ?
Answer (Detailed Solution Below)
Normal Subgroup and Quotient Groups Question 1 Detailed Solution
Explanation:
Option (1) is incorrect because normality does not imply that the subgroup is cyclic
Option (2) is incorrect because G/N is not necessarily abelian for all normal subgroups N
Option (3) is correct because the definition of a normal subgroup is \(g N g^{-1} = N \) for all \(g \in G \)
Option (4) is incorrect because a normal subgroup does not have to be trivial or the whole group;
normal subgroups can have other non-trivial forms
Hence Option(3) is the Correct Answer.
Normal Subgroup and Quotient Groups Question 2:
If F is homomorphism of a group G into another group G' with kernel k, then which of the following is true?
Answer (Detailed Solution Below)
Normal Subgroup and Quotient Groups Question 2 Detailed Solution
Explanation:
If F is homomorphism of a group G into another group G' with kernel k.
Since e ∈ k so k ≠ ϕ
Let a, b ∈ k
f(ab-1) = f(a)f(b-1)
= f(a)f(b)-1
= e1e1-1 = e1
So, ab-1 ∈ k
Hence k is a subgroup of G.
Let a ∈ G and h ∈ K
Then
f(aha-1) = f(a)f(h)f(a-1)
= f(a)f(h)f(a)-1
= f(a)e1f(a)-1 = e1
So, aha-1 ∈ k
So k is a normal subgroup of G.
Option (1) is true.
Top Normal Subgroup and Quotient Groups MCQ Objective Questions
Normal Subgroup and Quotient Groups Question 3:
Let G be a group and N a subgroup of G . Which of the following statements is true about N being a normal subgroup of G ?
Answer (Detailed Solution Below)
Normal Subgroup and Quotient Groups Question 3 Detailed Solution
Explanation:
Option (1) is incorrect because normality does not imply that the subgroup is cyclic
Option (2) is incorrect because G/N is not necessarily abelian for all normal subgroups N
Option (3) is correct because the definition of a normal subgroup is \(g N g^{-1} = N \) for all \(g \in G \)
Option (4) is incorrect because a normal subgroup does not have to be trivial or the whole group;
normal subgroups can have other non-trivial forms
Hence Option(3) is the Correct Answer.
Normal Subgroup and Quotient Groups Question 4:
If F is homomorphism of a group G into another group G' with kernel k, then which of the following is true?
Answer (Detailed Solution Below)
Normal Subgroup and Quotient Groups Question 4 Detailed Solution
Explanation:
If F is homomorphism of a group G into another group G' with kernel k.
Since e ∈ k so k ≠ ϕ
Let a, b ∈ k
f(ab-1) = f(a)f(b-1)
= f(a)f(b)-1
= e1e1-1 = e1
So, ab-1 ∈ k
Hence k is a subgroup of G.
Let a ∈ G and h ∈ K
Then
f(aha-1) = f(a)f(h)f(a-1)
= f(a)f(h)f(a)-1
= f(a)e1f(a)-1 = e1
So, aha-1 ∈ k
So k is a normal subgroup of G.
Option (1) is true.