Order of Automorphism Group of Prime Group

Theorem
Let $p$ be a prime number.

Let $G$ be a group of order $p$.

Let $\Aut G$ denote the automorphism group of $G$.

Then:
 * $\order {\Aut G} = p - 1$

where $\order {\, \cdot \,}$ denotes the order of a group.

Proof
From Prime Group is Cyclic we have that $G$ is a cyclic group.

From Order of Automorphism Group of Cyclic Group:
 * $\order {\Aut G} = \map \phi p$

where $\map \phi n$ denotes the Euler $\phi$ function.

The result follows from Euler Phi Function of Prime.