Group extension theory studies how a group \(G\) is assembled from a normal subgroup \(N\) and quotient \(Q\). It upgrades knowledge of adjacent normal-series factors to knowledge of how those factors interact. [@isaacs2008]
Definition
Two extensions are equivalent if an isomorphism of the middle groups induces the identity on the specified kernel and quotient. Abstract isomorphism of the middle groups alone is weaker.
Core result
Choose a lift of each element of \(Q\) and conjugate \(N\). Different lifts differ by inner automorphisms, giving a canonical outer action. A homomorphic section removes the multiplication defect and yields a semidirect product; a general section has a defect measured by a \(2\)-cocycle.
Structural properties
- The isomorphism types of kernel and quotient usually do not determine the middle group.
- Splitting is equivalent to a homomorphic section and to the existence of a complement to the kernel.
- A central extension has trivial quotient action on the kernel but may still be nonsplit.
- Finite coprime extensions split under Schur–Zassenhaus, with conjugacy of complements under solvability.
- Kaloujnine–Krasner embeds every extension in a suitable wreath product.
Extension theory explains why composition factors do not reconstruct a group: action and cocycle gluing data remain necessary.
Example and boundary
Knowledge network
The split-extension core article gives the multiplication formula, complements encode sections, and Schur–Zassenhaus handles finite coprime extensions.
Proof and sources
Isaacs, Chapter 3 organizes extensions through complements, semidirect and wreath products, and coprime action. Cohomological language precisely locates the general obstruction. [@isaacs2008]
