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Representations and Characters of Groups, Second Edition by Martin Liebeck, Gordon James

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5

FG-submodules and reducibility

We begin the study of FG-modules by introducing the basic building blocks of the theory – the irreducible FG-modules. First we require the notion of an FG-submodule of an FG-module.

Throughout, G is a group and F is image or image.

FG-submodules

5.1 Definition

Let V be an FG-module. A subset W of V is said to be an FG-submodule of V if W is a subspace and wgW for all wW and all gG.

Thus an FG-submodule of V is a subspace which is also an FG-module.

5.2 Examples

(1) For every FG-module V, the zero subspace {0}, and ...

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