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?

Question

Answer

Here, 6 = 2 x 3

Answer

Generator of a number n is less than n and relatively prime to n. so the generators are 1 and 5 answer is 2

Answer

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).

Answer

Use Euler's phi function as follows :
Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n and denoted by Greek letter phi as φ(n). The number of generators of a cyclic group can be found using Euler's phi function.
So here φ(6) = φ(3)×φ(2) [as 3 & 2 are relatively prime]. = 2.

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