IIT JAM Exam > IIT JAM Questions > Let G be a group of order 20 in which the con...
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Let G be a group of order 20 in which the conjugacy classes have sizes 1, 4, 5, 5, 5. Then which of the followings is/are TRUE ?
- a)G contains a normal subgroup of order 5.
- b)G contains a non- normal subgroup of order 5.
- c)G contains a subgroup of order 10.
- d)G contains a normal subgroup of order 4.
Correct answer is option 'A,C'. Can you explain this answer?
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Let G be a group of order 20 in which the conjugacy classes have sizes...
Solution:
Given, Group G has order 20 and the conjugacy classes have sizes 1, 4, 5, 5, 5.
Conjugacy Classes
- There is only one conjugacy class of size 1. Therefore, there exists an element of G which is the identity element. Let this element be denoted by e.
- There are three conjugacy classes of size 5. Therefore, there are three distinct elements of order 5 in G. Let these elements be denoted by a, b, c.
- There is one conjugacy class of size 4. Therefore, there are two distinct elements of order 2 in G. Let these elements be denoted by x, y.
Order of the Group
We know that the order of a group is equal to the sum of the squares of the sizes of its conjugacy classes. Therefore, we have:
1^2 + 4^2 + 5^2 + 5^2 + 5^2 = 1 + 16 + 25 + 25 + 25 = 92
Since the order of G is 20, this is a contradiction. Therefore, no such group G exists.
This means that the given question is incorrect and hence, has no solution.
Hence, the answer to this question is not possible.
Given, Group G has order 20 and the conjugacy classes have sizes 1, 4, 5, 5, 5.
Conjugacy Classes
- There is only one conjugacy class of size 1. Therefore, there exists an element of G which is the identity element. Let this element be denoted by e.
- There are three conjugacy classes of size 5. Therefore, there are three distinct elements of order 5 in G. Let these elements be denoted by a, b, c.
- There is one conjugacy class of size 4. Therefore, there are two distinct elements of order 2 in G. Let these elements be denoted by x, y.
Order of the Group
We know that the order of a group is equal to the sum of the squares of the sizes of its conjugacy classes. Therefore, we have:
1^2 + 4^2 + 5^2 + 5^2 + 5^2 = 1 + 16 + 25 + 25 + 25 = 92
Since the order of G is 20, this is a contradiction. Therefore, no such group G exists.
This means that the given question is incorrect and hence, has no solution.
Hence, the answer to this question is not possible.
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Let G be a group of order 20 in which the conjugacy classes have sizes 1, 4, 5, 5, 5. Then which of the followings is/are TRUE ?a)G contains a normal subgroup of order 5.b)G contains a non- normal subgroup of order 5.c)G contains a subgroup of order 10.d)G contains a normal subgroup of order 4.Correct answer is option 'A,C'. Can you explain this answer?
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Let G be a group of order 20 in which the conjugacy classes have sizes 1, 4, 5, 5, 5. Then which of the followings is/are TRUE ?a)G contains a normal subgroup of order 5.b)G contains a non- normal subgroup of order 5.c)G contains a subgroup of order 10.d)G contains a normal subgroup of order 4.Correct answer is option 'A,C'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Let G be a group of order 20 in which the conjugacy classes have sizes 1, 4, 5, 5, 5. Then which of the followings is/are TRUE ?a)G contains a normal subgroup of order 5.b)G contains a non- normal subgroup of order 5.c)G contains a subgroup of order 10.d)G contains a normal subgroup of order 4.Correct answer is option 'A,C'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let G be a group of order 20 in which the conjugacy classes have sizes 1, 4, 5, 5, 5. Then which of the followings is/are TRUE ?a)G contains a normal subgroup of order 5.b)G contains a non- normal subgroup of order 5.c)G contains a subgroup of order 10.d)G contains a normal subgroup of order 4.Correct answer is option 'A,C'. Can you explain this answer?.
Let G be a group of order 20 in which the conjugacy classes have sizes 1, 4, 5, 5, 5. Then which of the followings is/are TRUE ?a)G contains a normal subgroup of order 5.b)G contains a non- normal subgroup of order 5.c)G contains a subgroup of order 10.d)G contains a normal subgroup of order 4.Correct answer is option 'A,C'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Let G be a group of order 20 in which the conjugacy classes have sizes 1, 4, 5, 5, 5. Then which of the followings is/are TRUE ?a)G contains a normal subgroup of order 5.b)G contains a non- normal subgroup of order 5.c)G contains a subgroup of order 10.d)G contains a normal subgroup of order 4.Correct answer is option 'A,C'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let G be a group of order 20 in which the conjugacy classes have sizes 1, 4, 5, 5, 5. Then which of the followings is/are TRUE ?a)G contains a normal subgroup of order 5.b)G contains a non- normal subgroup of order 5.c)G contains a subgroup of order 10.d)G contains a normal subgroup of order 4.Correct answer is option 'A,C'. Can you explain this answer?.
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