---
date: 2022-04-05 23:42
modification date: Wednesday 13th April 2022 09:56:49
title: "Maschke's theorem"
aliases: ["Maschke's theorem", "Reynolds operator"]
---

Last modified date: <%+ tp.file.last_modified_date() %>

---
- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [representation theory](Unsorted/Representation%20Theory%20(Subject%20MOC).md)

---

# Maschke's theorem

![](attachments/Pasted%20image%2020220413095651.png)

Any submodule $V \leq W \in \modsleft{G}$ has a $G\dash$ invariant complement.
Proof: choose $\pi:W\to V$ a projection and define
$$
\pi_{G}(x)=\frac{1}{|G|} \sum_{g \in G} g \cdot \pi\left(g^{-1} \cdot x\right).
$$

Alternative statement: $kG \in \kalg$ is semisimple.