Heineken-Liebeck theorem
From Groupprops
Contents
Statement
Suppose is a finite group, and is a prime number. Then, there exists a finite -group if nilpotence class two and exponent such that the image of the natural map:
is isomorphic to .
Proof
Proof idea
The proof has two main parts:
- Given a finite group , construct a strongly connected finite directed graph such that is isomorphic to the automorphism group of . For full proof, refer: Every finite group is the automorphism group of a strongly connected finite directed graph
- Given a strongly connected directed graph and a prime number , construct a -group of class two and exponent such that the image of in has exponent dividing .
References
Textbook references=
- Finite Groups II by Bertram Huppert and Norman Blackburn, ISBN 0387106324 (English), ISBN 3540106324 (German), ^{More info}, Page 404, Section 13.6