Maschke's theorem

Let $G$ be a finite group and $k$ a field. Consider finite-dimensional representations of $G$ over $k$, equivalently finite-dimensional $kG$-modules.

Theorem (Maschke)

Assume that $|G|$ is invertible in $k$ (equivalently $\mathrm{char}(k)\nmid |G|$). Then every finite-dimensional $k$-representation $V$ of $G$ is semisimple: for every $G$-subrepresentation $W\subseteq V$, there exists a $G$-stable subspace $U\subseteq V$ such that

as $G$-representations (equivalently as $kG$-modules). In particular, every representation is completely reducible .

In module language: every finite-dimensional $kG$-module is a semisimple module .

Standard averaging proof (key construction)

Let $W\subseteq V$ be $G$-stable. Choose any $k$-linear projection $p:V\to W$ (not necessarily $G$-equivariant). Define the averaged map

Then:

• $p_G$ is $G$-equivariant, i.e. a module homomorphism $V\to W$,
• $p_G|_W = \mathrm{id}_W$, so $p_G$ is a $G$-equivariant projection,
• therefore $\ker(p_G)$ is $G$-stable and gives the complement $U=\ker(p_G)$.

This averaging step is exactly where the condition $\frac{1}{|G|}\in k$ is used.

Remarks and consequences

• Over $k=\mathbb C$, Maschke always applies to finite groups, yielding complete reducibility over \u211a .
• The theorem is equivalent to semisimplicity of the group algebra $k[G]$ (as a ring), and it underlies character theory (e.g. orthogonality of irreducible characters ).

Examples

1. A “good characteristic” decomposition for $S_3$ over $\mathbb C$.
   Let $S_3$ act on $\mathbb C^3$ by permuting coordinates (a permutation representation). The 1-dimensional subspace

   $$W = \{(a,a,a):a\in\mathbb C\}$$

   is $S_3$-stable (the trivial subrepresentation). Maschke guarantees an invariant complement, and indeed one is

   $$U=\{(x_1,x_2,x_3)\in\mathbb C^3 : x_1+x_2+x_3=0\},$$

   giving $\mathbb C^3 \cong W \oplus U$, where $U$ is the 2-dimensional standard irreducible.

2. A “good characteristic” example where not all irreducibles are 1D.
   Take $G=C_3=\langle g\rangle$ over $k=\mathbb F_2$. Since $\mathrm{char}(\mathbb F_2)=2\nmid 3$, Maschke applies: every $\mathbb F_2[C_3]$-module is semisimple.
   In particular, $\mathbb F_2[C_3]$ (the regular representation) decomposes as a direct sum of irreducibles, but over $\mathbb F_2$ there is a 2-dimensional irreducible factor (because $x^2+x+1$ is irreducible over $\mathbb F_2$). So semisimple does not mean “splits into 1-dimensional pieces”; it means “splits into irreducibles over the given field.”

3. Failure in “bad characteristic”: $C_p$ over $\mathbb F_p$.
   Let $G=C_p=\langle g\rangle$ and $k=\mathbb F_p$, so $\mathrm{char}(k)=p\mid |G|$. Consider the 2-dimensional representation $V=k^2$ where $g$ acts by the unipotent matrix

   $$\rho(g)=\begin{pmatrix}1&1\\0&1\end{pmatrix}.$$

   The line $W=\langle e_1\rangle$ is $G$-stable, but there is no $G$-stable complement line: the only eigenvectors of $\rho(g)$ lie in $\langle e_1\rangle$, so no other 1-dimensional subspace is preserved. Hence $V$ is not a direct sum of subrepresentations, and Maschke’s conclusion fails.