Ausgewählte Kapitel aus der Gruppentheorie: Σ-Theorie

A G-module clearly defines a ZG-module structure on A, and a ZG-module structure on A clearly defines a G-module structure on A. The distinction will be blurred in this regard, and one often just says that A is a G-module. The free G-modules are up to isomorphism ⊕ZG with basis X. For an arbitrary G-module A, there exists a free G-module F and a surjective map ε : F A, such that G ∼= F/ ker ε. This will be called a free presentation of the G-module A.