1. ## Cyclic groups.

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

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

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

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

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

Tonio