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.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $8$: Homomorphisms, Normal Subgroups and Quotient Groups: Exercise $27 \ \text {(i)}$