The Frobenius kernel consists of the identity together with all fixed-point-free elements in a Frobenius action. Closure is Frobenius's theorem, and nilpotence is Thompson's theorem. [@isaacs2008]
Definition
A point stabilizer \(H\) complements the kernel, with \(G=N\rtimes H\). The kernel acts regularly, so \(|N|=|\Omega|\).
Core result
The original existence proof uses characters; transfer provides another route. Nilpotence is split into solvable and general cases and uses Thompson's normal-complement theorem. See the Frobenius core article.
Structural properties
- \(|N|\equiv1\pmod{|H|}\), so kernel and complement orders are coprime.
- The kernel contains all fixed-point-free elements, which outside the identity form unions of conjugacy classes.
- A Frobenius kernel is nilpotent but need not be abelian.
- If the complement contains an involution, the kernel is abelian of odd order and the involution acts by inversion.
- Every \(H\)-invariant quotient of the kernel inherits a Frobenius action.
The kernel converts a permutation condition into canonical normal structure and makes every Frobenius group a coprime semidirect product.
Example and boundary
Knowledge network
The Frobenius core article proves existence and nilpotence, the complement article describes the fixed-point-free action, and nilpotent theory gives structural consequences.
Proof and sources
Recognition and centralizer criteria are Theorems 6.4-6.7; solvable and general nilpotence are Theorems 6.22-6.24. [@isaacs2008]
