# Maschke’s theorem

Let $G$ be a finite group^{}, and $k$ a field of characteristic^{} not dividing $|G|$. Then any representation $V$ of $G$ over $k$ is completely reducible.

###### Proof.

We need only show that any subrepresentation has a complement, and the result follows by induction^{}.

Let $V$ be a representation of $G$ and $W$ a subrepresentation. Let $\pi :V\to W$ be an arbitrary projection, and let

$${\pi}^{\prime}(v)=\frac{1}{|G|}\sum _{g\in G}{g}^{-1}\pi (gv)$$ |

This map is obviously $G$-equivariant, and is the identity^{} on $W$, and its image is contained in $W$, since $W$ is invariant under $G$. Thus it is an equivariant projection to $W$, and its kernel is a complement to $W$.
∎

Title | Maschke’s theorem |
---|---|

Canonical name | MaschkesTheorem |

Date of creation | 2013-03-22 13:21:16 |

Last modified on | 2013-03-22 13:21:16 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 9 |

Author | bwebste (988) |

Entry type | Theorem |

Classification | msc 20C15 |