# Heineken-Liebeck theorem

## Statement

Suppose $G$ is a finite group, and $p$ is a prime number. Then, there exists a finite $p$-group $P$ if nilpotence class two and exponent $p^2$ such that the image of the natural map:

$\operatorname{Aut}(P) \to \operatorname{Aut}(P/Z(P))$

is isomorphic to $G$.

## Proof

### Proof idea

The proof has two main parts:

1. Given a finite group $G$, construct a strongly connected finite directed graph $D$ such that $G$ is isomorphic to the automorphism group of $D$. For full proof, refer: Every finite group is the automorphism group of a strongly connected finite directed graph
2. Given a strongly connected directed graph $D$ and a prime number $p$, construct a $p$-group $P$ of class two and exponent $p^2$ such that the image of $\operatorname{Aut}(P)$ in $\operatorname{Aut}(P/Z(P))$ has exponent dividing $p^2$.