Finite Cyclic Group has Euler Phi Generators

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

Then $C_n$ has $\map \phi n$ generators, where $\map \phi n$ denotes the Euler $\phi$ function.

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

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

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