Maschke's theorem

The set of Maschke ( after Heinrich Maschke, 1899) is a central message from the mathematical branch of representation theory of finite groups. He states that representations are composed except in the special case of modular representations of irreducible representations.

Let be a finite group and a body. The essence of the theory of linear representations of depends fundamentally on whether the characteristics of a divisor of the order of or not. In the former case, one speaks of modular representations. The difference lies mainly in the statement of the theorem of Maschke founded.

Non- modular case

It is true; This is particularly the case when characteristic has 0, so for example.

Then the set of states Maschke:

Each - linear representation of is a direct sum of irreducible representations.

Equivalent formulations are:

• Each representation is semisimple.
• Every invariant subspace of a representation has an invariant complement, ie.

Modular representations

Applies the other hand, the following applies: The group ring is not completely reducible, ie the regular representation is not completely reducible.

Not everyone has a sub-module of complement.