A crossed homomorphism is an ordinary homomorphism law twisted by a group action. Its graph is precisely a complement in a semidirect product, turning complement classification into a functional equation. [@isaacs2008]
Definition
Crossed homomorphisms are also called \(1\)-cocycles. For \(n\in N\), the map \(\delta_n(h)=n^{-1}{}^h n\) is principal, or a \(1\)-coboundary.
Core result
Multiply graph elements: \((\delta(h),h)(\delta(k),k)=(\delta(h){}^h\delta(k),hk)\). Closure is exactly the cocycle identity. Conjugating the graph by \((n,1)\) produces the principal-coboundary change.
Structural properties
- For abelian \(N\), crossed homomorphisms form an abelian group \(Z^1(H,N)\).
- Quotienting by principal cocycles gives \(H^1(H,N)\); in the nonabelian case it is generally only a pointed set.
- A complement corresponds to a cocycle graph, with the standard complement represented by the zero cocycle.
- Under the solvable coprime hypotheses of Schur–Zassenhaus, conjugacy of complements says that the relevant \(H^1\) is trivial.
- The map \(h\mapsto g^{-1}g^h\) is the basic principal crossed homomorphism.
Crossed homomorphisms are common coordinates for extensions, affine actions, and nonabelian cohomology, preserving the noncommutative order imposed by the action.
Example and boundary
Knowledge network
Complements are cocycle graphs, split extensions provide the semidirect environment, and coprime action determines when all complements are conjugate.
Proof and sources
Graph and conjugacy parametrization come from Sections 3A-3B. Cohomological notation summarizes the same calculation and does not replace its group-theoretic proof. [@isaacs2008]
