Maschke's Theorem

Theorem 1 (Maschke's Theorem): If $G$ is a finite group then every group representation of $G$ is completely reducible.

• Proof: We prove this by induction on the dimension of a group representation of $G$.

• First suppose that $(V, \rho)$ is a $1$-dimensional group representation of $G$. Then $V$ is trivially completely reducible.

• So suppose that for some $n \in \mathbb{N}$, $n > 1$ we have that any $n$-dimensional group representation of $G$ is completely reducible. Let $(V, \rho)$ be an $n + 1$-dimensional group representation of $G$.

• Since $G$ is a finite group, we have by Weyl's Unitarity Trick that $(V, \rho)$ can be assumed to be a unitary representation of $G$. By the theorem on the Unitary Group Representations page we have that $(V, \rho)$ is either irreducible or decomposable.

• If $(V, \rho)$ is irreducible then we are done. So suppose that $(V, \rho)$ is decomposable. Then $V \cong U \oplus W$ where $U$ and $W$ are nonzero proper subrepresentations of $V$. In particular, $\mathrm{dim}(U), \mathrm{dim}(W) \leq n$. So by the induction hypothesis, $U$ and $W$ are both completely reducible and we can write $U \cong U_1 \oplus U_2 \oplus ... \oplus U_s$ and $W \cong W_1 \oplus W_2 \oplus ... \oplus W_t$, where $U_1, U_2, ..., U_s, W_1, W_2, ..., W_t$ are irreducible subrepresentations of $U$ and $W$, and thus of $V$. So:

(1)

\begin{align} \quad V \cong [U_1 \oplus U_2 \oplus ... \oplus U_s] \oplus [W \cong W_1 \oplus W_2 \oplus ... \oplus W_t] \end{align}

• So $V$ is completely reducible.

• By the principle of mathematical induction, we see that for any $n \in \mathbb{N}$, every $n$-dimensional representation of the finite group $G$ is completely reducible. $\blacksquare$