1. Cyclic groups.

Let $G$ be a cyclic groups. (Exist $x\in G$ so that $G=={x^n:n\in \mathbb{Z}$)

Show that if $ord(x)=n$ then the number of genereted elements of $$ is $\phi(n)$.

2. Originally Posted by Also sprach Zarathustra
Let $G$ be a cyclic groups. (Exist $x\in G$ so that $G=={x^n:n\in \mathbb{Z}$)

Show that if $ord(x)=n$ then the number of genereted elements of $$ is $\phi(n)$.

Hint: show that $x^k$ is a generator of the group iff $gcd(k,n)=1$ , using Euclides algorithm.

Tonio