5,365 views
1 votes
1 votes
How to find no. of generators of cyclic group of orden n ?

1 Answer

Best answer
6 votes
6 votes

Let (G,*) be a Cyclic group of order ' n ': The number of Generators is G="Φ(n)" 

Euler's totient function counts the positive integers up to a given integer n that are relatively prime (co- prime) to n. 

 Co-prime : It can be defined more formally as the number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd(nk) is equal to 1. 

For eg: 

Number of generators of cyclic group of order 3  =  Φ(3) ={1,2} = 2 generators .

Number of generators of cyclic group of order 7  =  Φ(7) = {1,2,3,4,5,6} = 6 generators .

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.

  •  n = p * q
  • Φ(n) = Φ(p) * Φ(q)

for example:  if we need to find out how many generators exists in cyclic group of order 77 then

                   77 = 7 * 11

                Φ(77) = Φ(7)  *  Φ(11)

By above explanation, Φ(7) = 6 generators  and  Φ(11) = 10 generators.

So total number of generators will be = 6 * 10 = 60 generators in cyclic group of order 77.

------------------------------------------------------------------------------------------------------------------

Eg 2: Number of generators in cyclic group of order 35:

           Φ(35) = Φ(7) * Φ(5)

                    = 6 * 4 =24 generators.

------------------------------------------------------------------------------------------------------------------

Another special cases: Number of generators in cyclic group of order 25:

                               Φ(25) = Φ(52)

General Formula is:   if Φ(Pn) = Pn - Pn-1

                           Now Φ(25) = 52 - 5(2-1)

                                          = 20 generators.

------------------------------------------------------------------------------------------------------------------

Eg:  Number of generators in cyclic group of order 84:

                      84 = 22 * 3 * 7

                   Φ(84) = Φ(22 * 3 * 7)

                            = Φ(22) * Φ(3) * Φ(7)

                            = 22 - 2(2-1) * 2 * 6

                             = 24 Generators

selected by

Related questions

5 votes
5 votes
1 answer
1
Lakshman Bhaiya asked Oct 7, 2018
2,112 views
Suppose that $G$ is a cyclic group of order $10$ with generator $a\in G$.Order of $a^{8}$ is _______
1 votes
1 votes
1 answer
2
ankitgupta.1729 asked Dec 6, 2017
1,134 views
Find all the subgroups of a cyclic group of order 12.(A) {e},(a6),(a4),(a3),(a2),(a)(B) (a12),(a6),(a4),(a3),(a2),(a)(C) (a12),(a6),(a4),(a2),(a)(D) (a12),(a6),(a4),(a3),...
1 votes
1 votes
1 answer
3
monty asked Sep 24, 2015
1,172 views
which of the following is wrong?A)group of order is 27 is cyclicB)group of order is 14 is cyclicC)group of order is 21 is cyclic  group of order is 30 is cyclic
3 votes
3 votes
2 answers
4
pratikb asked Dec 23, 2014
2,534 views
If G is an infinite cyclic group then which of the following is not true?a) G has exactly 2 generators.b) G is isomorphic to (Z,+).c) Every proper sub group of G is fini...