We now come to our first major result in representation theory, namely Maschke’s Theorem. A consequence of this theorem is that every FG-module is a direct sum of irreducible FG-submodules, where as usual F = or . (The assumption on F is important – see Example 8.2(2) below.) This essentially reduces representation theory to the study of irreducible FG-modules.
8.1 Maschke’s Theorem
Let G be a finite group, let F be or , and let V be an FG-module. If U is an FG-submodule of V, then there is an FG-submodule ...