A normal \(p\)-complement collects the entire \(p'\)-part of a finite group into a normal subgroup, making the group a split extension by a Sylow \(p\)-subgroup. Its existence is the central global-separation problem in \(p\)-local theory. [@isaacs2008]
Definition
For \(P\in Syl_p(G)\), Schur–Zassenhaus gives \(G=N\rtimes P\). The subgroup \(O^p(G)\) is the intersection of all normal subgroups of \(p\)-power index; when a normal complement exists, \(O^p(G)=N\).
Core result
A common strategy first proves that fusion or transfer is controlled by a Sylow normalizer, then obtains a sufficiently large abelian \(p\)-quotient, and finally uses Tate or local induction to upgrade to the full \(p\)-residual. The linked articles contain the proofs.
Structural properties
- A normal \(p\)-complement is unique, since it is the largest normal \(p'\)-subgroup.
- It exists exactly when \(O^p(G)\) is a \(p'\)-group.
- Inheritance by arbitrary subgroups is not automatic without checking intersections and quotient structure.
- A nilpotent group has a normal \(p\)-complement for every prime; conversely, having one for every prime forces nilpotence.
- Nilpotence of a Frobenius kernel is proved by constructing normal \(p\)-complements prime by prime.
The phrase does not ask whether a complement to a Sylow \(p\)-subgroup merely exists; it requires the \(p'\)-complement itself to be normal in the whole group.
Example and boundary
Knowledge network
Burnside and Frobenius supply local criteria, Thompson subgroups compress the tests, and transfer detects the abelianized obstruction.
Proof and sources
Definitions and classical criteria run through Chapters 5-7; this supporting article centralizes terminology and links each proof. [@isaacs2008]
