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Number of elements of the cyclic group of order 6 can be used as generators of the group are
- a)= 3
- b)= 5
- c)= 2
- d)4
Correct answer is option 'C'. Can you explain this answer?
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Number of elements of the cyclic group of order 6 can be used as gener...
Here, 6 = 2 x 3
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Number of elements of the cyclic group of order 6 can be used as gener...
Generator of a number n is less than n and relatively prime to n. so the generators are 1 and 5 answer is 2
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Number of elements of the cyclic group of order 6 can be used as gener...
To find the number of elements that can be generators of a cyclic group of order 6, we first need to understand what a cyclic group is and how it operates.
Cyclic Group:
A cyclic group is a group that is generated by a single element. This means that all the elements in the group can be obtained by repeatedly applying the group operation to the generator. In other words, a cyclic group is a group that is generated by a single element, and every element in the group can be expressed as a power of the generator.
Order of an Element:
The order of an element in a group is the smallest positive integer n such that raising the element to the power of n gives the identity element of the group. In a cyclic group, the order of the generator is equal to the order of the whole group.
Finding the Generators:
To find the number of elements that can be generators of a cyclic group of order 6, we need to determine the order of each element in the group.
Let's consider a cyclic group of order 6. The possible orders of the elements in the group can be 1, 2, 3, 6.
1. Order 1:
The identity element of the group has order 1. However, the identity element cannot be a generator because it does not generate any other elements.
2. Order 2:
If an element has order 2, it means that squaring the element gives the identity element. In a cyclic group of order 6, there can be only one element of order 2. This is because if there were more than one element of order 2, it would contradict the fact that the group is cyclic. Therefore, there is only one element of order 2, which is not a generator.
3. Order 3:
If an element has order 3, it means that cubing the element gives the identity element. In a cyclic group of order 6, there can be two elements of order 3. These two elements are generators of the group.
4. Order 6:
The generator of a cyclic group of order 6 has order 6 itself. There can be only one element of order 6 in the group, which is the generator.
Therefore, the number of elements that can be generators of the cyclic group of order 6 is 2 (corresponding to the elements of order 3 and 6).
Hence, the correct answer is option 'C' (2).
Cyclic Group:
A cyclic group is a group that is generated by a single element. This means that all the elements in the group can be obtained by repeatedly applying the group operation to the generator. In other words, a cyclic group is a group that is generated by a single element, and every element in the group can be expressed as a power of the generator.
Order of an Element:
The order of an element in a group is the smallest positive integer n such that raising the element to the power of n gives the identity element of the group. In a cyclic group, the order of the generator is equal to the order of the whole group.
Finding the Generators:
To find the number of elements that can be generators of a cyclic group of order 6, we need to determine the order of each element in the group.
Let's consider a cyclic group of order 6. The possible orders of the elements in the group can be 1, 2, 3, 6.
1. Order 1:
The identity element of the group has order 1. However, the identity element cannot be a generator because it does not generate any other elements.
2. Order 2:
If an element has order 2, it means that squaring the element gives the identity element. In a cyclic group of order 6, there can be only one element of order 2. This is because if there were more than one element of order 2, it would contradict the fact that the group is cyclic. Therefore, there is only one element of order 2, which is not a generator.
3. Order 3:
If an element has order 3, it means that cubing the element gives the identity element. In a cyclic group of order 6, there can be two elements of order 3. These two elements are generators of the group.
4. Order 6:
The generator of a cyclic group of order 6 has order 6 itself. There can be only one element of order 6 in the group, which is the generator.
Therefore, the number of elements that can be generators of the cyclic group of order 6 is 2 (corresponding to the elements of order 3 and 6).
Hence, the correct answer is option 'C' (2).
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Number of elements of the cyclic group of order 6 can be used as generators of the group area)= 3b)= 5c)= 2d)4Correct answer is option 'C'. Can you explain this answer?
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Number of elements of the cyclic group of order 6 can be used as generators of the group area)= 3b)= 5c)= 2d)4Correct 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 Number of elements of the cyclic group of order 6 can be used as generators of the group area)= 3b)= 5c)= 2d)4Correct 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 Number of elements of the cyclic group of order 6 can be used as generators of the group area)= 3b)= 5c)= 2d)4Correct answer is option 'C'. Can you explain this answer?.
Number of elements of the cyclic group of order 6 can be used as generators of the group area)= 3b)= 5c)= 2d)4Correct 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 Number of elements of the cyclic group of order 6 can be used as generators of the group area)= 3b)= 5c)= 2d)4Correct 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 Number of elements of the cyclic group of order 6 can be used as generators of the group area)= 3b)= 5c)= 2d)4Correct answer is option 'C'. Can you explain this answer?.
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