Heineken-Liebeck theorem

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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.

References

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