A Frobenius complement is a point stabilizer and a complement to the kernel. Its fixed-point-free action on nonidentity kernel elements imposes much stronger structural restrictions than order alone. [@isaacs2008]
Definition
The conjugates of \(H\) are the distinct point stabilizers, and their nonidentity parts are pairwise disjoint.
Core result
Fixed-point-free coprime action excludes \(C_p\times C_p\), so a Sylow \(p\)-subgroup has at most one subgroup of order \(p\). The finite p-group classification lemma yields cyclic or generalized quaternion. See Theorems 6.9-6.19.
Structural properties
- \(|H|\) divides \(|N|-1\), so complement and kernel orders are coprime.
- Every subgroup of a complement remains fixed-point-free on the kernel and is itself a Frobenius complement.
- If \(|H|\) is even, \(H\) has a unique involution and the kernel is abelian.
- The complement as a whole need not be cyclic, abelian, or nilpotent.
- Its action on the kernel is faithful; only the identity centralizes the whole kernel.
The Sylow restrictions arise because all local pieces must act fixed-point-freely on one common kernel, not merely from the order of the complement.
Example and boundary
Knowledge network
The kernel supplies the action target, fixed-point-free automorphisms explain the local condition, and generalized quaternion groups are the Sylow-\(2\) exception.
Proof and sources
Equivalent criteria and structural restrictions are Theorems 6.4 and 6.9-6.19; detailed derivations are in the Frobenius core article. [@isaacs2008]
