Finding the Number of Generators in a Cyclic Group

A cyclic group is a group that can be generated by a single element. This element is called a generator of the group. If a group \(G\) is cyclic and has order \(n\), it means that there exists an element \(g \in G\) such that every element in \(G\) can be written as a power of \(g\), i.e., \(G = \{g^0, g^1, g^2, \dots, g^{n-1}\}\), and \(g^n\) is the identity element.

The question asks for the number of generators in a cyclic group of order 10. In a finite cyclic group of order \(n\), the elements that are generators are precisely those elements \(g^k\) (where \(g\) is one generator) such that \(\gcd(k, n) = 1\). The number of such integers \(k\) with \(1 \le k \le n\) and \(\gcd(k, n) = 1\) is given by Euler's totient function, denoted by \(\phi(n)\).

Using Euler's Totient Function for Cyclic Groups

Euler's totient function, \(\phi(n)\), counts the number of positive integers up to a given integer \(n\) that are relatively prime to \(n\). For a cyclic group of order \(n\), the number of generators is exactly \(\phi(n)\).

In this case, the order of the cyclic group is \(n=10\). We need to calculate \(\phi(10)\).

To calculate \(\phi(10)\), we first find the prime factorization of 10:

\(10 = 2 \times 5\)

Since 2 and 5 are distinct prime numbers, we can use the property of Euler's totient function that for distinct primes \(p_1\) and \(p_2\), \(\phi(p_1 p_2) = \phi(p_1) \phi(p_2)\). Also, for a prime number \(p\), \(\phi(p) = p-1\).

So, \(\phi(10) = \phi(2 \times 5) = \phi(2) \times \phi(5)\).

• \(\phi(2) = 2 - 1 = 1\)
• \(\phi(5) = 5 - 1 = 4\)

Therefore, the number of generators is:

\(\phi(10) = 1 \times 4 = 4\)

Conclusion on the Number of Generators

The number of generators in a cyclic group of order 10 is 4. These generators correspond to the elements \(g^k\) where \(g\) is a generator and \(k\) is a positive integer less than or equal to 10 such that \(\gcd(k, 10)=1\). The values of \(k\) that satisfy this condition are 1, 3, 7, and 9. There are exactly \(\phi(10)=4\) such values of \(k\), leading to 4 distinct generators.

Thus, the number of generators in a cyclic group of order 10 is 4.