## How many generators does a cyclic group of order n have?

Number of generators of cyclic group of order 6 = Φ(6) ={1,5} = 2 generators . Suppose if the number is large then what will u do : If n is very large then we need to do, split the n in such a way that it becomes multiplication of two prime numbers. By above explanation, Φ(7) = 6 generators and Φ(11) = 10 generators.

## What is the order of generator of a cyclic group?

A cyclic group is a group which is equal to one of its cyclic subgroups: G = ⟨g⟩ for some element g, called a generator. For a finite cyclic group G of order n we have G = {e, g, g2, , gn−1}, where e is the identity element and gi = gj whenever i ≡ j (mod n); in particular gn = g0 = e, and g−1 = gn−1.

How do you find the number of generators in a cyclic group?

An element am ∈ G is also a generator of G is HCF of m and 8 is 1. HCF of 1 and 8 is 1, HCF of 3 and 8 is 1, HCF of 5 and 8 is 1, HCF of 7 and 8 is 1. Hence, a, a3, a5, a7 are generators of G. Therefore, there are four generators of G.

### What are the generators of cyclic group of order 10?

More Groups Questions

• Q1. The multiplicative group {1, -1, i, -i} is a cyclic group, its generators are.
• Q2. The number of generators of the cyclic group G of order 8 is.
• Q3. The number of generators of a cyclic group of order 10 is.
• Q4.
• Q5.
• Q6.
• Q7.
• Q8.

### How many generators are there of the cyclic group of order 6?

(c) How many elements of a cyclic group of order n are generators for that group? |〈1〉| = 1 |〈x〉| = 6 |〈x2〉| = 3 |〈x3〉| = 2 |〈x4〉| = 3 |〈x5〉| = 6. Therefore there are only two elements of G which generate G. As described above, they are precisely the elements of order 6.

How many generators are there of the cyclic group G of order 10?

Hence there are four generators of G. Similarly you can find generators of groups of order 10, 12, 6 etc.

#### How many generators are there in a cyclic group of order 60?

No of generators in Group (Cyclic group) too is given by Euler’s_totient_function, i.e. no of elements less than N & Co prime to N. No of generators possible are =60(1−1/2)(1−1/3)(1−1/5)=60∗1/2∗2/3∗4/5=16. So total 16 Generators !

#### How many generators does a cyclic group of order 36 have?

How many generators are there in the cyclic group Z36? Generators of this group are numbers that are coprime to 36. That is, the generators are {1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35}, so there are 12 generators.

How many generators are there of the cyclic group G of order 6?

Therefore there are only two elements of G which generate G. As described above, they are precisely the elements of order 6.

## How many generators does a cyclic group of order 12 have?

4
The number of generators of a cyclic group of order 12 is ________. Correct answer is ‘4’.

## How many generators a cyclic group of order 60 has?

How do you prove cyclic groups are generators of order n?

Suppose G is a cyclic group of order n, then there is at least one g ∈ G such that the order of g equals n, that is: g n = e and g k ≠ e for 0 ≤ k < n. Let us prove that the elements of the following set { g s | 0 ≤ s < n, gcd (s, n) = 1 } are all generators of G.

### Is every cyclic group isomorphic to a generator?

Every cyclic group is isomorphic to either Z or Z / n Z if it is infinite or finite. If it is infinite, it’ll have generators ± 1. If it is finite of order n, any element of the group with order relatively prime to n is a generator.

### What is the value of GCD of a cyclic group of order n?

A cyclic group of order n has exactly φ ( n) generators where φ is Euler’s totient function. gcd ( k, n) = 1. φ ( n) = n ∏ p ∣ n ( 1 − 1 p). Show activity on this post.

What is not an example of a cyclic group?

Non-example of cyclic groups: Klein’s 4-group is a group of order 4. It is not a cyclic group. Let (G, ∘) be a cyclic group generated by a.