Maschke's Theorem

Theorem

$V$ is a representation of a finite group $G$ over a field $\mathbb{F}$ with characteristic not dividing the order of $G$. If $V$ has a sub-representation $W$, then it has another sub-representation $U$ such that $V=W\oplus U$.

Maschke's theorem

Every representation of a finite group $G$ over a field $\mathbb{F}$ with characteristic not dividing the order of $G$ is a direct sum of irreducible representations.

Theorem

Let $\phi :G\to \mathrm{GL}(V)$ be a unitary representation of a group. Then $\phi$ is either irreducible or decomposable.

\begin{proof} See Representation theory of finite groups - 2012 - Steinberg.pdf, proposition 3.2.3. \end{proof}