GATE Exam > GATE Questions > Consider the following groups(1) G1 = ( {1, 3...
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Consider the following groups
(1) G1 = ( {1, 3, 5, 7}, ⊗8) , where ⊗8 is multiplication modulo 8 operation.
(2) G2 = ( {1, 2, 3, 4, 5, 6}, ⊗7), where ⊗7is multiplication modulo 7 operation.
(3) G3 = ( {0, 1, 2, 3, 4},⊗5)where ⊗5) is addition modulo 5 operation.
(4) G4 = ( {1, 2, 4, 5, 7, 8}, ⊗9), where ⊗9 is multiplication modulo 9 operation.
The number of Cyclic groups is ___________________?
- a)4
- b)3
- c)2
- d)1
Correct answer is option 'B'. Can you explain this answer?
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Consider the following groups(1) G1 = ( {1, 3, 5, 7}, ⊗8) , whe...
A group (G, *) is called a cyclic group if there exists an element a∈G, such that every element of ‘G’ can be written an for some integer n. Then ‘a’ is called a generating element / generator.
G1 do not have any generators. So, it is not a cyclic group.
G2 have 3 and 5 as generators.
G3 have 1, 2, 3 and 4 as generators.
G4 also has a generator. One generator is enough to say whether a group is cyclic or not. Here 2 is the generator.
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Consider the following groups(1) G1 = ( {1, 3, 5, 7}, ⊗8) , whe...
To determine if G1 is a group, we need to check if it satisfies the four group axioms:
1. Closure: For any two elements a, b in G1, the product a * b is also in G1.
Since G1 is defined as the set {1, 3, 5, 7}, we can check the product of every pair of elements in this set:
1 * 1 = 1 -> 1 is in G1
1 * 3 = 3 -> 3 is in G1
1 * 5 = 5 -> 5 is in G1
1 * 7 = 7 -> 7 is in G1
3 * 1 = 3 -> 3 is in G1
3 * 3 = 9 -> 9 is not in G1
3 * 5 = 15 -> 15 is not in G1
3 * 7 = 21 -> 21 is not in G1
5 * 1 = 5 -> 5 is in G1
5 * 3 = 15 -> 15 is not in G1
5 * 5 = 25 -> 25 is not in G1
5 * 7 = 35 -> 35 is not in G1
7 * 1 = 7 -> 7 is in G1
7 * 3 = 21 -> 21 is not in G1
7 * 5 = 35 -> 35 is not in G1
7 * 7 = 49 -> 49 is not in G1
As we can see, not all possible products of elements in G1 are also in G1. Therefore, G1 does not satisfy the closure axiom and is not a group.
Note: It is important to mention that the definition of the group operation (e.g., addition, multiplication, etc.) was not given in the question. However, regardless of the specific operation, if closure is not satisfied, then the set cannot form a group.
1. Closure: For any two elements a, b in G1, the product a * b is also in G1.
Since G1 is defined as the set {1, 3, 5, 7}, we can check the product of every pair of elements in this set:
1 * 1 = 1 -> 1 is in G1
1 * 3 = 3 -> 3 is in G1
1 * 5 = 5 -> 5 is in G1
1 * 7 = 7 -> 7 is in G1
3 * 1 = 3 -> 3 is in G1
3 * 3 = 9 -> 9 is not in G1
3 * 5 = 15 -> 15 is not in G1
3 * 7 = 21 -> 21 is not in G1
5 * 1 = 5 -> 5 is in G1
5 * 3 = 15 -> 15 is not in G1
5 * 5 = 25 -> 25 is not in G1
5 * 7 = 35 -> 35 is not in G1
7 * 1 = 7 -> 7 is in G1
7 * 3 = 21 -> 21 is not in G1
7 * 5 = 35 -> 35 is not in G1
7 * 7 = 49 -> 49 is not in G1
As we can see, not all possible products of elements in G1 are also in G1. Therefore, G1 does not satisfy the closure axiom and is not a group.
Note: It is important to mention that the definition of the group operation (e.g., addition, multiplication, etc.) was not given in the question. However, regardless of the specific operation, if closure is not satisfied, then the set cannot form a group.
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Consider the following groups(1) G1 = ( {1, 3, 5, 7}, ⊗8) , where ⊗8 is multiplication modulo 8 operation.(2) G2 = ( {1, 2, 3, 4, 5, 6}, ⊗7), where ⊗7is multiplication modulo 7 operation.(3) G3 = ( {0, 1, 2, 3, 4},⊗5)where ⊗5) is addition modulo 5 operation.(4) G4 = ( {1, 2, 4, 5, 7, 8}, ⊗9), where ⊗9 is multiplication modulo 9 operation.The number of Cyclic groups is ___________________?a)4b)3c)2d)1Correct answer is option 'B'. Can you explain this answer?
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Consider the following groups(1) G1 = ( {1, 3, 5, 7}, ⊗8) , where ⊗8 is multiplication modulo 8 operation.(2) G2 = ( {1, 2, 3, 4, 5, 6}, ⊗7), where ⊗7is multiplication modulo 7 operation.(3) G3 = ( {0, 1, 2, 3, 4},⊗5)where ⊗5) is addition modulo 5 operation.(4) G4 = ( {1, 2, 4, 5, 7, 8}, ⊗9), where ⊗9 is multiplication modulo 9 operation.The number of Cyclic groups is ___________________?a)4b)3c)2d)1Correct answer is option 'B'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Consider the following groups(1) G1 = ( {1, 3, 5, 7}, ⊗8) , where ⊗8 is multiplication modulo 8 operation.(2) G2 = ( {1, 2, 3, 4, 5, 6}, ⊗7), where ⊗7is multiplication modulo 7 operation.(3) G3 = ( {0, 1, 2, 3, 4},⊗5)where ⊗5) is addition modulo 5 operation.(4) G4 = ( {1, 2, 4, 5, 7, 8}, ⊗9), where ⊗9 is multiplication modulo 9 operation.The number of Cyclic groups is ___________________?a)4b)3c)2d)1Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following groups(1) G1 = ( {1, 3, 5, 7}, ⊗8) , where ⊗8 is multiplication modulo 8 operation.(2) G2 = ( {1, 2, 3, 4, 5, 6}, ⊗7), where ⊗7is multiplication modulo 7 operation.(3) G3 = ( {0, 1, 2, 3, 4},⊗5)where ⊗5) is addition modulo 5 operation.(4) G4 = ( {1, 2, 4, 5, 7, 8}, ⊗9), where ⊗9 is multiplication modulo 9 operation.The number of Cyclic groups is ___________________?a)4b)3c)2d)1Correct answer is option 'B'. Can you explain this answer?.
Consider the following groups(1) G1 = ( {1, 3, 5, 7}, ⊗8) , where ⊗8 is multiplication modulo 8 operation.(2) G2 = ( {1, 2, 3, 4, 5, 6}, ⊗7), where ⊗7is multiplication modulo 7 operation.(3) G3 = ( {0, 1, 2, 3, 4},⊗5)where ⊗5) is addition modulo 5 operation.(4) G4 = ( {1, 2, 4, 5, 7, 8}, ⊗9), where ⊗9 is multiplication modulo 9 operation.The number of Cyclic groups is ___________________?a)4b)3c)2d)1Correct answer is option 'B'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Consider the following groups(1) G1 = ( {1, 3, 5, 7}, ⊗8) , where ⊗8 is multiplication modulo 8 operation.(2) G2 = ( {1, 2, 3, 4, 5, 6}, ⊗7), where ⊗7is multiplication modulo 7 operation.(3) G3 = ( {0, 1, 2, 3, 4},⊗5)where ⊗5) is addition modulo 5 operation.(4) G4 = ( {1, 2, 4, 5, 7, 8}, ⊗9), where ⊗9 is multiplication modulo 9 operation.The number of Cyclic groups is ___________________?a)4b)3c)2d)1Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following groups(1) G1 = ( {1, 3, 5, 7}, ⊗8) , where ⊗8 is multiplication modulo 8 operation.(2) G2 = ( {1, 2, 3, 4, 5, 6}, ⊗7), where ⊗7is multiplication modulo 7 operation.(3) G3 = ( {0, 1, 2, 3, 4},⊗5)where ⊗5) is addition modulo 5 operation.(4) G4 = ( {1, 2, 4, 5, 7, 8}, ⊗9), where ⊗9 is multiplication modulo 9 operation.The number of Cyclic groups is ___________________?a)4b)3c)2d)1Correct answer is option 'B'. Can you explain this answer?.
Solutions for Consider the following groups(1) G1 = ( {1, 3, 5, 7}, ⊗8) , where ⊗8 is multiplication modulo 8 operation.(2) G2 = ( {1, 2, 3, 4, 5, 6}, ⊗7), where ⊗7is multiplication modulo 7 operation.(3) G3 = ( {0, 1, 2, 3, 4},⊗5)where ⊗5) is addition modulo 5 operation.(4) G4 = ( {1, 2, 4, 5, 7, 8}, ⊗9), where ⊗9 is multiplication modulo 9 operation.The number of Cyclic groups is ___________________?a)4b)3c)2d)1Correct answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for GATE.
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Consider the following groups(1) G1 = ( {1, 3, 5, 7}, ⊗8) , where ⊗8 is multiplication modulo 8 operation.(2) G2 = ( {1, 2, 3, 4, 5, 6}, ⊗7), where ⊗7is multiplication modulo 7 operation.(3) G3 = ( {0, 1, 2, 3, 4},⊗5)where ⊗5) is addition modulo 5 operation.(4) G4 = ( {1, 2, 4, 5, 7, 8}, ⊗9), where ⊗9 is multiplication modulo 9 operation.The number of Cyclic groups is ___________________?a)4b)3c)2d)1Correct answer is option 'B'. Can you explain this answer?, a detailed solution for Consider the following groups(1) G1 = ( {1, 3, 5, 7}, ⊗8) , where ⊗8 is multiplication modulo 8 operation.(2) G2 = ( {1, 2, 3, 4, 5, 6}, ⊗7), where ⊗7is multiplication modulo 7 operation.(3) G3 = ( {0, 1, 2, 3, 4},⊗5)where ⊗5) is addition modulo 5 operation.(4) G4 = ( {1, 2, 4, 5, 7, 8}, ⊗9), where ⊗9 is multiplication modulo 9 operation.The number of Cyclic groups is ___________________?a)4b)3c)2d)1Correct answer is option 'B'. Can you explain this answer? has been provided alongside types of Consider the following groups(1) G1 = ( {1, 3, 5, 7}, ⊗8) , where ⊗8 is multiplication modulo 8 operation.(2) G2 = ( {1, 2, 3, 4, 5, 6}, ⊗7), where ⊗7is multiplication modulo 7 operation.(3) G3 = ( {0, 1, 2, 3, 4},⊗5)where ⊗5) is addition modulo 5 operation.(4) G4 = ( {1, 2, 4, 5, 7, 8}, ⊗9), where ⊗9 is multiplication modulo 9 operation.The number of Cyclic groups is ___________________?a)4b)3c)2d)1Correct answer is option 'B'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Consider the following groups(1) G1 = ( {1, 3, 5, 7}, ⊗8) , where ⊗8 is multiplication modulo 8 operation.(2) G2 = ( {1, 2, 3, 4, 5, 6}, ⊗7), where ⊗7is multiplication modulo 7 operation.(3) G3 = ( {0, 1, 2, 3, 4},⊗5)where ⊗5) is addition modulo 5 operation.(4) G4 = ( {1, 2, 4, 5, 7, 8}, ⊗9), where ⊗9 is multiplication modulo 9 operation.The number of Cyclic groups is ___________________?a)4b)3c)2d)1Correct answer is option 'B'. Can you explain this answer? tests, examples and also practice GATE tests.
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