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Maschke's theorem for finite groups over fields whose characteristic does not divide $∣ G ∣$

Statement

Let $G$ be a finite group, let $k$ be a field with $char k ∤ ∣ G ∣$ , let $ρ : G \to GL (V)$ be a finite-dimensional representation of $G$ over $k$ , and let $W \leq V$ be a subrepresentation. Then there is a subrepresentation $U \leq V$ such that

$V = W \oplus U .$

Facts & Assumptions

Given: A finite group $G$ , a field $k$ with $char k ∤ ∣ G ∣$ , a finite-dimensional representation $ρ : G \to GL (V)$ , and a subrepresentation $W \leq V$ .

[L1]

A subrepresentation is a linear subspace stable under every group element (Subrepresentations, direct sums of representations, and irreducibility).

[A1]

Because $char k ∤ ∣ G ∣$ , the scalar $∣ G ∣ \cdot 1_{k}$ is nonzero in $k$ and therefore has a multiplicative inverse, denoted $∣ G ∣^{- 1}$ .

[A2]

Since $V$ is finite-dimensional over $k$ , the subspace $W$ has a $k$ -linear complement, so there is a $k$ -linear projection $P : V \to W$ with $P (w) = w$ for every $w \in W$ .

Proof

technique · direct

1.1A1A2givenconstruct

Choose the projection $P$ from [A2] and define $P_{G} : = ∣ G ∣^{- 1} \sum_{g \in G} ρ (g) P ρ (g)^{- 1} .$ Each summand is $k$ -linear, so $P_{G}$ is a $k$ -linear endomorphism of $V$ .

2.1step 1.1givenalgebra

For every $h \in G$ , $ρ (h) P_{G} ρ (h)^{- 1} = ∣ G ∣^{- 1} \sum_{g \in G} ρ (h g) P ρ (h g)^{- 1} = P_{G},$ because left multiplication by $h$ permutes the finite set $G$ . Thus $P_{G}$ is $G$ -equivariant.

3.1step 1.1step 2.1L1givenalgebra

Each summand $ρ (g) P ρ (g)^{- 1}$ maps $V$ into $ρ (g) (W) = W$ , so $P_{G} (V) \subseteq W$ by [L1]. If $w \in W$ , then $ρ (g)^{- 1} w \in W$ by [L1], hence $P (ρ (g)^{- 1} w) = ρ (g)^{- 1} w$ and therefore $ρ (g) P ρ (g)^{- 1} (w) = w$ for every $g$ . Summing gives $P_{G} (w) = w$ . So $P_{G}$ has image exactly $W$ .

4.1step 2.1step 3.1L1givenalgebra∎

Put $U : = \ker P_{G}$ . Since $P_{G}$ is $G$ -equivariant, $U$ is a subrepresentation. For every $v \in V$ one has $v = P_{G} (v) + (v - P_{G} (v)),$ with $P_{G} (v) \in W$ and $v - P_{G} (v) \in U$ . If $x \in W \cap U$ , then step 3.1 gives $x = P_{G} (x) = 0$ , so the sum is direct. Hence $V = W \oplus U$ .

Depends on

Used by

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Peter Webb, A Course in Finite Group Representation Theory, Theorem 1.2.1 (standard reference, not scraped)
Pavel Etingof et al., Introduction to Representation Theory, Theorem 3.1 (standard reference, not scraped)

Maschke's theorem for finite groups over fields whose characteristic does not divide ∣G∣∣G∣