An automorphism is fixed-point-free when it fixes only the identity. In finite groups this is extremely restrictive: the acting and target orders are coprime and the target acquires strong solvable or nilpotent structure. [@isaacs2008]
Definition
The identity is necessarily fixed; 'fixed-point-free' means no nonidentity fixed point.
Core result
\(A\) is semiregular on \(N\setminus\{1\}\), so every orbit has size \(|A|\) and \(|A|\mid|N|-1\). The semidirect product satisfies the Frobenius intersection criterion. Kernel nilpotence is proved in the core article.
Structural properties
- A fixed-point-free automorphism of prime order makes a finite group nilpotent, with sharper results bounding class.
- If its order is \(2\), the finite group is abelian of odd order and the automorphism is inversion.
- The property descends to invariant quotients; a subgroup must be invariant before restriction makes sense.
- On an abelian group it says that the linear operator \(a-1\) is invertible.
- For an acting group, every nonidentity element must be checked; checking generators alone is insufficient.
This turns a dynamical fixed-point condition into Frobenius semidirect structure and is the most rigid extreme of coprime action.
Example and boundary
Knowledge network
Frobenius groups externalize the action as a semidirect product, coprime action supplies general tools, and nilpotent theory describes Thompson's conclusion.
Proof and sources
Orbit counting, coprimeness, semidirect criteria, and kernel nilpotence are in Sections 6A-6C. [@isaacs2008]
