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If G be a group of order p q r , p < q < r all three no. are primes then which of the following is/are true?
- a)G has a normal subgroup of order r.
- b)G has a normal subgroup of order qr.
- c)G has a normal subgroup of order q , where qX( r - 1)
- d)G has a normal subgroup of order p
Correct answer is option 'A,B,C'. Can you explain this answer?
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Verified Answer
If G be a group of order p q r , p < q < r all three no. are pri...
It is a very well known theorem that " If G be a group o f order p q r ,
p < q < r being primes , then all the subgroup of order r is normal in G.
Hence option (A) is correct.
Again group G has a normal subgroup of order pq .
and , if q + (r -1) then G has a normal subgroup of order q.
But it is not necessary that the subgroup of order p is normal, it may be or not.
Most Upvoted Answer
If G be a group of order p q r , p < q < r all three no. are pri...
By Sylow's theorems, we know that the number of Sylow p-subgroups, denoted by n_p, divides the order of G and is congruent to 1 modulo p. Similarly, n_q divides the order of G and is congruent to 1 modulo q, and n_r divides the order of G and is congruent to 1 modulo r.
Since p, q, and r are distinct primes, we can conclude that n_p, n_q, and n_r are all equal to 1. This means that G has a unique Sylow p-subgroup, a unique Sylow q-subgroup, and a unique Sylow r-subgroup.
Since the number of Sylow p-subgroups is 1, the Sylow p-subgroup must be normal in G. Similarly, the Sylow q-subgroup and Sylow r-subgroup are also normal in G.
Now, consider the subgroup H = Sylow p-subgroup * Sylow q-subgroup * Sylow r-subgroup. Since all three subgroups are normal in G, their product H is a subgroup of G. Moreover, the order of H is p*q*r, which is the order of G.
Therefore, H = G, and we can conclude that G is the direct product of its Sylow p, q, and r-subgroups.
Since p, q, and r are distinct primes, we can conclude that n_p, n_q, and n_r are all equal to 1. This means that G has a unique Sylow p-subgroup, a unique Sylow q-subgroup, and a unique Sylow r-subgroup.
Since the number of Sylow p-subgroups is 1, the Sylow p-subgroup must be normal in G. Similarly, the Sylow q-subgroup and Sylow r-subgroup are also normal in G.
Now, consider the subgroup H = Sylow p-subgroup * Sylow q-subgroup * Sylow r-subgroup. Since all three subgroups are normal in G, their product H is a subgroup of G. Moreover, the order of H is p*q*r, which is the order of G.
Therefore, H = G, and we can conclude that G is the direct product of its Sylow p, q, and r-subgroups.
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If G be a group of order p q r , p < q < r all three no. are pri...
It is a very well known theorem that " If G be a group o f order p q r ,
p < q < r being primes , then all the subgroup of order r is normal in G.
Hence option (A) is correct.
Again group G has a normal subgroup of order pq .
and , if q + (r -1) then G has a normal subgroup of order q.
But it is not necessary that the subgroup of order p is normal, it may be or not.
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If G be a group of order p q r , p < q < r all three no. are primes then which of the following is/are true?a)G has a normal subgroup of order r.b)G has a normal subgroup of order qr.c)G has a normal subgroup of order q , where qX( r - 1)d)G has a normal subgroup of order pCorrect answer is option 'A,B,C'. Can you explain this answer?
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If G be a group of order p q r , p < q < r all three no. are primes then which of the following is/are true?a)G has a normal subgroup of order r.b)G has a normal subgroup of order qr.c)G has a normal subgroup of order q , where qX( r - 1)d)G has a normal subgroup of order pCorrect answer is option 'A,B,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 If G be a group of order p q r , p < q < r all three no. are primes then which of the following is/are true?a)G has a normal subgroup of order r.b)G has a normal subgroup of order qr.c)G has a normal subgroup of order q , where qX( r - 1)d)G has a normal subgroup of order pCorrect answer is option 'A,B,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 If G be a group of order p q r , p < q < r all three no. are primes then which of the following is/are true?a)G has a normal subgroup of order r.b)G has a normal subgroup of order qr.c)G has a normal subgroup of order q , where qX( r - 1)d)G has a normal subgroup of order pCorrect answer is option 'A,B,C'. Can you explain this answer?.
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ample number of questions to practice If G be a group of order p q r , p < q < r all three no. are primes then which of the following is/are true?a)G has a normal subgroup of order r.b)G has a normal subgroup of order qr.c)G has a normal subgroup of order q , where qX( r - 1)d)G has a normal subgroup of order pCorrect answer is option 'A,B,C'. Can you explain this answer? tests, examples and also practice Mathematics tests.
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