# Maschke's theorem

In mathematics, **Maschke's theorem**,[1][2] named after Heinrich Maschke,[3] is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces. Maschke's theorem allow one to make general conclusions about representations of a finite group *G* without actually computing them. It reduces the task of classifying all representations to a more manageable task of classifying irreducible representations, since when the theorem applies, any representation is a direct sum of irreducible pieces (constituents). Moreover, it follows from the Jordan–Hölder theorem that, while the decomposition into a direct sum of irreducible subrepresentations may not be unique, the irreducible pieces have well-defined multiplicities. In particular, a representation of a finite group over a field of characteristic zero is determined up to isomorphism by its character.

## Formulations

Maschke's theorem addresses the question: when is a general (finite-dimensional) representation built from irreducible subrepresentations using the direct sum operation? This question (and its answer) are formulated differently for different perspectives on group representation theory.

### Group-theoretic

Maschke's theorem is commonly formulated as a corollary to the following result:

Theorem.IfVis a complex representation of a finite groupGwith a subrepresentationW, then there is another subrepresentationUofVsuch thatV=W⊕U.[4][5]

Then the corollary is

Corollary (Maschke's theorem).Every representation of a finite groupGover a fieldFwith characteristic not dividing the order ofGis a direct sum of irreducible representations.[6][7]

The vector space of complex-valued class functions of a group `G` has a natural `G`-invariant inner product structure, described in the article Schur orthogonality relations. Maschke's theorem was originally proved for the case of representations over by constructing `U` as the orthogonal complement of `W` under this inner product.

### Module-theoretic

One of the approaches to representations of finite groups is through module theory. *Representations* of a group *G* are replaced by *modules* over its group algebra *K*[*G*] (to be precise, there is an isomorphism of categories between ** K[G]-Mod** and

**Rep**, the category of representations of

_{G}*G*). Irreducible representations correspond to simple modules. In the module-theoretic language, Maschke's theorem asks: is an arbitrary module semisimple? In this context, the theorem can be reformulated as follows:

Maschke's Theorem.LetGbe a finite group andKa field whose characteristic does not divide the order ofG. ThenK[G], the group algebra ofG, is semisimple.[8][9]

The importance of this result stems from the well developed theory of semisimple rings, in particular, the Artin–Wedderburn theorem (sometimes referred to as Wedderburn's Structure Theorem). When *K* is the field of complex numbers, this shows that the algebra *K*[*G*] is a product of several copies of complex matrix algebras, one for each irreducible representation.[10] If the field *K* has characteristic zero, but is not algebraically closed, for example, *K* is a field of real or rational numbers, then a somewhat more complicated statement holds: the group algebra *K*[*G*] is a product of matrix algebras over division rings over *K*. The summands correspond to irreducible representations of *G* over *K*.[11]

### Category-theoretic

Reformulated in the language of semi-simple categories, Maschke's theorem states

Maschke's theorem.IfGis a group andFis a field with characteristic not dividing the order ofG, then the category of representations ofGoverFis semi-simple.

## Proofs

### Module-theoretic

Let *V* be a *K*[*G*]-submodule. We will prove that *V* is a direct summand. Let *π* be any *K*-linear projection of *K*[*G*] onto *V*. Consider
the map given by
Then *φ* is again a projection: it is clearly *K*-linear, maps *K*[*G*] onto *V*, and induces the identity on *V*. Moreover we have

so *φ* is in fact *K*[*G*]-linear. By the splitting lemma, . This proves that every submodule is a direct summand, that is, *K*[*G*] is semisimple.

## Converse statement

The above proof depends on the fact that #*G* is invertible in *K*. This might lead one to ask if the converse of Maschke's theorem also holds: if the characteristic of *K* divides the order of *G*, does it follow that *K*[*G*] is not semisimple? The answer is *yes*.[12]

**Proof.** For define . Let . Then *I* is a *K*[*G*]-submodule. We will prove that for every nontrivial submodule *V* of *K*[*G*], . Let *V* be given, and let be any nonzero element of *V*. If , the claim is immediate. Otherwise, let . Then so and so that is an element of both *I* and *V*. This proves that *V* is not a direct complement of *I* for all *V*, so *K*[*G*] is not semisimple.

## Notes

- Maschke, Heinrich (1898-07-22). "Ueber den arithmetischen Charakter der Coefficienten der Substitutionen endlicher linearer Substitutionsgruppen" [On the arithmetical character of the coefficients of the substitutions of finite linear substitution groups].
*Math. Ann.*(in German).**50**(4): 492–498. doi:10.1007/BF01444297. JFM 29.0114.03. MR 1511011. - Maschke, Heinrich (1899-07-27). "Beweis des Satzes, dass diejenigen endlichen linearen Substitutionsgruppen, in welchen einige durchgehends verschwindende Coefficienten auftreten, intransitiv sind" [Proof of the theorem that those finite linear substitution groups, in which some everywhere vanishing coefficients appear, are intransitive].
*Math. Ann.*(in German).**52**(2–3): 363–368. doi:10.1007/BF01476165. JFM 30.0131.01. MR 1511061. - O'Connor, John J.; Robertson, Edmund F., "Heinrich Maschke",
*MacTutor History of Mathematics archive*, University of St Andrews. - Fulton & Harris, Proposition 1.5.
- Serre, Theorem 1.
- Fulton & Harris, Corollary 1.6.
- Serre, Theorem 2.
- It follows that every module over
*K*[*G*] is a semisimple module. - The converse statement also holds: if the characteristic of the field divides the order of the group (the
*modular case*), then the group algebra is not semisimple. - The number of the summands can be computed, and turns out to be equal to the number of the conjugacy classes of the group.
- One must be careful, since a representation may decompose differently over different fields: a representation may be irreducible over the real numbers but not over the complex numbers.
- Serre, Exercise 6.1.

## References

- Lang, Serge (2002-01-08).
*Algebra*. Graduate Texts in Mathematics,**211**(Revised 3rd ed.). New York: Springer-Verlag. ISBN 978-0-387-95385-4. MR 1878556. Zbl 0984.00001. - Serre, Jean-Pierre (1977-09-01).
*Linear Representations of Finite Groups*. Graduate Texts in Mathematics,**42**. New York–Heidelberg: Springer-Verlag. ISBN 978-0-387-90190-9. MR 0450380. Zbl 0355.20006. - Fulton, William; Harris, Joe (1991).
*Representation theory. A first course*. Graduate Texts in Mathematics, Readings in Mathematics.**129**. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.