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Choose the correct answer. If His a normal subgroup of G and K is a normal subgroup of H, then
- a)K is a normal subgroup of G
- b)K is not a normal subgroup of G
- c)K is a subgroup of G.
- d)K is a subgroup of G
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Choose the correct answer. If His a normal subgroup of G and K is a no...
Let G = A4, H = {(1 2)(3 4),(1 3)(2 4),(1 4)(2 3), e},
K = {e, (1 2) (3 4)}
Then H is normal in G and K is normal in H. But K is not normal in G
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Choose the correct answer. If His a normal subgroup of G and K is a no...
Let G = A4, H = {(1 2)(3 4),(1 3)(2 4),(1 4)(2 3), e},
K = {e, (1 2) (3 4)}
Then H is normal in G and K is normal in H. But K is not normal in G
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Community Answer
Choose the correct answer. If His a normal subgroup of G and K is a no...
Explanation:
To determine whether K is a normal subgroup of G, we need to examine the properties of normal subgroups and apply them to the given information.
Definition of a normal subgroup:
A subgroup H of a group G is said to be a normal subgroup if for every element g in G, the conjugate of H by g, denoted by gHg⁻¹, is also a subgroup of G.
Given information:
1. H is a normal subgroup of G
2. K is a normal subgroup of H
Consequence:
Since K is a subgroup of H (given information) and H is a normal subgroup of G (given information), it does not necessarily mean that K is a normal subgroup of G. Therefore, option 'B' is the correct answer.
Explanation:
1. It is given that H is a normal subgroup of G, which means that for every element g in G, the conjugate of H by g, denoted by gHg⁻¹, is also a subgroup of G.
2. It is also given that K is a normal subgroup of H, which means that for every element h in H, the conjugate of K by h, denoted by hKh⁻¹, is also a subgroup of H.
Counterexample:
To show that K may not be a normal subgroup of G, we can provide a counterexample.
Consider the group G = S₃, the symmetric group of order 3, and let H = {e, (12)}, the subgroup generated by the permutation (12).
Since H = {e, (12)} is a subgroup of G and it has only two elements, it is also a normal subgroup of G.
Now, let K = {e}, the trivial subgroup. K is a subgroup of H and is also a normal subgroup of H.
However, K = {e} is not a normal subgroup of G because for the element (23) in G, the conjugate of K by (23) is {(23)} which is not a subgroup of G.
Conclusion:
From the given information, it cannot be concluded that K is a normal subgroup of G. Therefore, the correct answer is option 'B' - K is not a normal subgroup of G.
To determine whether K is a normal subgroup of G, we need to examine the properties of normal subgroups and apply them to the given information.
Definition of a normal subgroup:
A subgroup H of a group G is said to be a normal subgroup if for every element g in G, the conjugate of H by g, denoted by gHg⁻¹, is also a subgroup of G.
Given information:
1. H is a normal subgroup of G
2. K is a normal subgroup of H
Consequence:
Since K is a subgroup of H (given information) and H is a normal subgroup of G (given information), it does not necessarily mean that K is a normal subgroup of G. Therefore, option 'B' is the correct answer.
Explanation:
1. It is given that H is a normal subgroup of G, which means that for every element g in G, the conjugate of H by g, denoted by gHg⁻¹, is also a subgroup of G.
2. It is also given that K is a normal subgroup of H, which means that for every element h in H, the conjugate of K by h, denoted by hKh⁻¹, is also a subgroup of H.
Counterexample:
To show that K may not be a normal subgroup of G, we can provide a counterexample.
Consider the group G = S₃, the symmetric group of order 3, and let H = {e, (12)}, the subgroup generated by the permutation (12).
Since H = {e, (12)} is a subgroup of G and it has only two elements, it is also a normal subgroup of G.
Now, let K = {e}, the trivial subgroup. K is a subgroup of H and is also a normal subgroup of H.
However, K = {e} is not a normal subgroup of G because for the element (23) in G, the conjugate of K by (23) is {(23)} which is not a subgroup of G.
Conclusion:
From the given information, it cannot be concluded that K is a normal subgroup of G. Therefore, the correct answer is option 'B' - K is not a normal subgroup of G.
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Choose the correct answer. If His a normal subgroup of G and K is a normal subgroup of H, thena)K is a normal subgroup of Gb)K is not a normal subgroup of Gc)K is a subgroup of G.d)K is a subgroup of GCorrect answer is option 'B'. Can you explain this answer?
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Choose the correct answer. If His a normal subgroup of G and K is a normal subgroup of H, thena)K is a normal subgroup of Gb)K is not a normal subgroup of Gc)K is a subgroup of G.d)K is a subgroup of GCorrect answer is option 'B'. 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 Choose the correct answer. If His a normal subgroup of G and K is a normal subgroup of H, thena)K is a normal subgroup of Gb)K is not a normal subgroup of Gc)K is a subgroup of G.d)K is a subgroup of GCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Choose the correct answer. If His a normal subgroup of G and K is a normal subgroup of H, thena)K is a normal subgroup of Gb)K is not a normal subgroup of Gc)K is a subgroup of G.d)K is a subgroup of GCorrect answer is option 'B'. Can you explain this answer?.
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Solutions for Choose the correct answer. If His a normal subgroup of G and K is a normal subgroup of H, thena)K is a normal subgroup of Gb)K is not a normal subgroup of Gc)K is a subgroup of G.d)K is a subgroup of GCorrect answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for Mathematics.
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