6 - Extensions of groups and modules

Summary

Chapter 6 considers various questions about extensions of groups and modules, most particularly the conjugacy of complements to some fixed normal subgroup in a split group extension. Suppose G is represented on an abelian group or F-space V and form the semidirect product GV. Section 17 shows there is a bijection between the set of conjugacy classes of complements to V in GV and the 1-cohomology group H¹(G, V). If V is an F-space so is H¹(G, V). Moreover if C_V(G) = 0 there is a largest member of the class of FG-modules U such that C_U(G) = 0 and U is the extension of V by a module centralized by G. Indeed it turns out that if U is the largest member of this class then U/V ≅ H¹(G, V). Further the dual of the statement is also true: that is if V = [V, G] then there is a largest FG-module U* such that U* = [U*, G] and U* is the extension of an FG-module Z by V with G centralizing Z. In this case Z ≅ H¹(G, V*).

These results together with Maschke's Theorem are then used to prove the Schur–Zassenhaus Theorem, which gives reasonably complete information about extensions of a finite group B by a finite group A when the orders of A and B are relatively prime. The Schur–Zassenhaus Theorem is then used to prove Phillip Hall's extended Sylow Theorem for solvable groups.