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Suppose N is a normal subgroup of a group G. Which one of the following is true?
- a)If G is an infinite group then G/N is an infinite group
- b)If G is a nonabelian group then G/N is a nonabelian group
- c)If G is a cyclic group then G/N is an abelian group
- d)If G is an abelian group then G/N is a cyclic group
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
Suppose N is a normal subgroup of a group G. Which one of the followin...
Explanation:
To prove that option 'C' is true, we need to show that if G is a cyclic group, then G/N is an abelian group.
Definition:
A group G is said to be cyclic if there exists an element g in G such that every element of G can be written as a power of g.
Proof:
Let G be a cyclic group and let g be a generator of G. This means that every element of G can be written as g^n, where n is an integer.
Now consider the quotient group G/N, where N is a normal subgroup of G. The elements of G/N are the cosets of N in G, denoted by gN for all g in G.
Claim 1:
For any two elements g_1N and g_2N in G/N, their product is commutative.
Proof of Claim 1:
Let g_1N and g_2N be two elements of G/N. This means that g_1N = g_1N and g_2N = g_2N for some g_1, g_2 in G.
Then, (g_1N)(g_2N) = (g_1g_2)N = (g_2g_1)N = (g_2N)(g_1N)
Therefore, the product of any two elements in G/N is commutative.
Claim 2:
The identity element of G/N is N.
Proof of Claim 2:
The identity element of G/N is the coset eN, where e is the identity element of G.
Since N is a normal subgroup, we have eN = N.
Therefore, the identity element of G/N is N.
Claim 3:
For any element gN in G/N, its inverse is (g^-1)N.
Proof of Claim 3:
Let gN be an element of G/N. This means that gN = gN for some g in G.
The inverse of gN is the coset (g^-1)N, where g^-1 is the inverse of g in G.
Since N is a normal subgroup, we have (g^-1)N = (g^-1)N.
Therefore, the inverse of gN is (g^-1)N.
Conclusion:
From the above claims, we can conclude that G/N is an abelian group when G is a cyclic group.
Therefore, option 'C' is true.
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Community Answer
Suppose N is a normal subgroup of a group G. Which one of the followin...
If any group is a cyclic group then it is Abelian group also. Here G, is a CYCLIC group.
We know that Quotient group(G/N) of CYCLIC group is a CYCLIC & ABELIAN group is ABELIAN.
Now, Quotient group (G/N) is CYCLIC group {since G is cyclic} implies it must be an ABELIAN group.
Therefore, Quotient group (G/N) is an ABELIAN group. So, option 'C' is correct.
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Suppose N is a normal subgroup of a group G. Which one of the following is true?a)If G is an infinite group then G/N is an infinite groupb)If G is a nonabelian group then G/N is a nonabelian groupc)If G is a cyclic group then G/N is an abelian groupd)If G is an abelian group then G/N is a cyclic groupCorrect answer is option 'C'. Can you explain this answer?
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Suppose N is a normal subgroup of a group G. Which one of the following is true?a)If G is an infinite group then G/N is an infinite groupb)If G is a nonabelian group then G/N is a nonabelian groupc)If G is a cyclic group then G/N is an abelian groupd)If G is an abelian group then G/N is a cyclic groupCorrect answer is option 'C'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Suppose N is a normal subgroup of a group G. Which one of the following is true?a)If G is an infinite group then G/N is an infinite groupb)If G is a nonabelian group then G/N is a nonabelian groupc)If G is a cyclic group then G/N is an abelian groupd)If G is an abelian group then G/N is a cyclic groupCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Suppose N is a normal subgroup of a group G. Which one of the following is true?a)If G is an infinite group then G/N is an infinite groupb)If G is a nonabelian group then G/N is a nonabelian groupc)If G is a cyclic group then G/N is an abelian groupd)If G is an abelian group then G/N is a cyclic groupCorrect answer is option 'C'. Can you explain this answer?.
Suppose N is a normal subgroup of a group G. Which one of the following is true?a)If G is an infinite group then G/N is an infinite groupb)If G is a nonabelian group then G/N is a nonabelian groupc)If G is a cyclic group then G/N is an abelian groupd)If G is an abelian group then G/N is a cyclic groupCorrect answer is option 'C'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Suppose N is a normal subgroup of a group G. Which one of the following is true?a)If G is an infinite group then G/N is an infinite groupb)If G is a nonabelian group then G/N is a nonabelian groupc)If G is a cyclic group then G/N is an abelian groupd)If G is an abelian group then G/N is a cyclic groupCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Suppose N is a normal subgroup of a group G. Which one of the following is true?a)If G is an infinite group then G/N is an infinite groupb)If G is a nonabelian group then G/N is a nonabelian groupc)If G is a cyclic group then G/N is an abelian groupd)If G is an abelian group then G/N is a cyclic groupCorrect answer is option 'C'. Can you explain this answer?.
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