Maschke's Theorem

Maschke's Theorem is one of the cornerstones of representation theory, establishing that all representations of finite groups in characteristic zero are completely reducible.

TheoremMaschke's Theorem

Let $G$ be a finite group and $\mathbb{F}$ a field such that $\text{char}(\mathbb{F}) \nmid |G|$ (in particular, if $\text{char}(\mathbb{F}) = 0$ ). Then every finite-dimensional representation of $G$ over $\mathbb{F}$ is completely reducible.

Equivalently, every subrepresentation has a complementary subrepresentation.

Proof Strategy: Given a subrepresentation $W \subseteq V$ , choose any vector space projection $\pi_0: V \to W$ . Average it over the group: $\pi = \frac{1}{|G|} \sum_{g \in G} \rho(g) \circ \pi_0 \circ \rho(g^{-1})$

This produces a $G$ -equivariant projection onto $W$ , whose kernel is a complementary subrepresentation.

ExampleApplications

Classification of representations: For finite groups over $\mathbb{C}$ , we need only classify irreducible representations. Every representation is a direct sum of irreducibles.

Character theory: The character of any representation is a linear combination of irreducible characters with non-negative integer coefficients.

Regular representation: For $G$ finite, the regular representation $\mathbb{C}[G]$ decomposes as: $\mathbb{C}[G] \cong \bigoplus_{V \text{ irreducible}} V^{\oplus \dim V}$

Remark

The hypothesis $\text{char}(\mathbb{F}) \nmid |G|$ is essential. When $\text{char}(\mathbb{F})$ divides $|G|$ (modular representation theory), complete reducibility fails spectacularly.

Counterexample: Let $G = \mathbb{Z}/p\mathbb{Z}$ over $\mathbb{F}_p$ . Consider $\mathbb{F}_p^2$ with generator $g$ acting as $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ . The span of $(1,0)$ is invariant, but has no complement—the representation is indecomposable but not irreducible.

TheoremConverse (Partial)

If every representation of $G$ over $\mathbb{F}$ is completely reducible, then either:

$\text{char}(\mathbb{F}) = 0$ , or
$\text{char}(\mathbb{F}) = p > 0$ and $p \nmid |G|$

In other words, the characteristic condition is necessary for complete reducibility.

DefinitionSemisimple Group Algebra

The group algebra $\mathbb{F}[G]$ is called semisimple if every $\mathbb{F}[G]$ -module is completely reducible.

Maschke's theorem states that $\mathbb{F}[G]$ is semisimple if and only if $\text{char}(\mathbb{F}) \nmid |G|$ .

For semisimple algebras, the Artin-Wedderburn theorem gives an isomorphism: $\mathbb{F}[G] \cong \prod_{i=1}^k \text{Mat}_{n_i}(\mathbb{F})$ where the product runs over irreducible representations.

Maschke's theorem transforms the study of finite group representations from a potentially intractable problem into a manageable one: classify irreducibles, compute their tensor products, and understand multiplicities. This foundation supports character theory, Frobenius reciprocity, and connections to number theory and physics.