Finite Cyclic Group has Euler Phi Generators

Theorem
Let $C_n$ be a (finite) cyclic group of order $n$.

Then $C_n$ has $\phi \left({n}\right)$ generators, where $\phi \left({n}\right)$ denotes the Euler $\phi$ function.

Proof
From List of Elements in Finite Cyclic Group, the elements of $G$ are:
 * $\left\{ {g^k: g \in G, 0 \le k < n}\right\}$

From Element is Generator of Cyclic Group iff Coprime with Order, $g^k$ generates $G$ iff $k \perp n$.

The result follows by definition of the Euler $\phi$ function.