A complement is the internal form of a split extension. It realizes the quotient as an actual subgroup and makes the assembly of kernel and quotient visible through conjugation. [@isaacs2008]
Definition
The quotient map \(\pi:G\to G/N\) restricts to an isomorphism \(H\cong G/N\). Every element has a unique expression \(nh\), although \(hn\) need not equal \(nh\).
Core result
The quotient restriction to a complement is an isomorphism and its inverse gives a section. Conversely, the image of a section is a complement. The semidirect isomorphism is \((n,h)\mapsto nh\). Coprime existence and conjugacy are proved in the core theorem article.
Structural properties
- A complement has order \(|G:N|\); coprimeness with \(|N|\) is an additional condition, not part of the definition.
- Complements need not be equal. Their conjugacy classes are described by crossed homomorphisms or nonabelian \(H^1\).
- If the complement is also normal, then \([N,H]\leq N\cap H=1\) and \(G=N\times H\).
- Existence of a complement is not the same as a normal complement. A \(p\)-complement has \(p'\)-order and \(p\)-power index; a normal \(p\)-complement must also be normal.
- In a Frobenius group, a point stabilizer complements the kernel and acts fixed-point-freely on it.
A complement lifts an abstract quotient to a concrete subgroup of \(G\), making conjugation, normalizers, and actions directly available.
Example and boundary
Knowledge network
Split extensions give the equivalent formulations, Schur–Zassenhaus supplies coprime existence and conjugacy, and crossed homomorphisms parametrize complements in a fixed semidirect product.
Proof and sources
Definition, section equivalence, and conjugacy questions are in Sections 3A-3B; full proofs are in the split-extension and Schur–Zassenhaus core articles. [@isaacs2008]
