# 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. If (Vρ) is a finite-dimensional representation of a finite group G over a field of characteristic zero, and U is an invariant subspace of V, then the theorem claims that U admits an invariant direct complement W; in other words, the representation (Vρ) is completely reducible. More generally, the theorem holds for fields of positive characteristic p, such as the finite fields, if the prime p does not divide the order of G.

## Reformulation and the meaning

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]. Irreducible representations correspond to simple modules. Maschke's theorem addresses the question: is a general (finite-dimensional) representation built from irreducible subrepresentations using the direct sum operation? In the module-theoretic language, is an arbitrary module semisimple? In this context, the theorem can be reformulated as follows:

Maschke's Theorem. Let G be a finite group and K a field whose characteristic does not divide the order of G. Then K[G], the group algebra of G, is semisimple.[4][5]

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.[6] 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.[7]

Returning to representation theory, Maschke's theorem and its module-theoretic version allow one to make general conclusions about representations of a finite group G without actually computing them. They reduce 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.

## Proof

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 ${\displaystyle \varphi :K[G]\to V}$ given by ${\displaystyle \varphi (x)={\frac {1}{\#G}}\sum _{s\in G}s\cdot \pi (s^{-1}\cdot x).}$ 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

{\displaystyle {\begin{aligned}\varphi (t\cdot x)&={\frac {1}{\#G}}\sum _{s\in G}s\cdot \pi (s^{-1}\cdot t\cdot x)\\{}&={\frac {1}{\#G}}\sum _{u\in G}t\cdot u\cdot \pi (u^{-1}\cdot x)\\{}&=t\cdot \varphi (x),\end{aligned}}}

so φ is in fact K[G]-linear. By the splitting lemma, ${\displaystyle K[G]=V\oplus \ker \varphi }$. 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.

Proof. For ${\displaystyle x=\sum \lambda _{g}g\in K[G]}$ define ${\displaystyle \epsilon (x)=\sum \lambda _{g}}$. Let ${\displaystyle I=\ker \epsilon }$. Then I is a K[G]-submodule. We will prove that for every nontrivial submodule V of K[G], ${\displaystyle I\cap V\neq 0}$. Let V be given, and let ${\displaystyle v=\sum \mu _{g}g}$ be any nonzero element of V. If ${\displaystyle \epsilon (v)=0}$, the claim is immediate. Otherwise, let ${\displaystyle s=\sum 1g}$. Then ${\displaystyle \epsilon (s)=\#G\cdot 1=0}$ so ${\displaystyle s\in I}$ and ${\displaystyle sv=\left(\sum 1g\right)\left(\sum \mu _{g}g\right)=\sum \epsilon (v)g=\epsilon (v)s}$ so that ${\displaystyle sv}$ 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

1. ^ 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. JFM 29.0114.03. MR 1511011. doi:10.1007/BF01444297.
2. ^ 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. JFM 30.0131.01. MR 1511061. doi:10.1007/BF01476165.
3. ^
4. ^ It follows that every module over K[G] is a semisimple module.
5. ^ 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.
6. ^ The number of the summands can be computed, and turns out to be equal to the number of the conjugacy classes of the group.
7. ^ 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.