PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-29

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A representation with kernel containing a normal subgroup factors through the quotient, and irreducibility is unchanged by inflation

Statement

Let $G$ be a group, let $N ⊴ G$ be a normal subgroup (Normal subgroup: invariance under conjugation), and let $ρ : G \to GL (V)$ be a representation over a field $k$ with $N \subseteq \ker ρ$ . Then:

the formula $\overline{ρ} (g N) : = ρ (g)$ defines a well-posed representation $\overline{ρ} : G / N \to GL (V)$ with $ρ = \overline{ρ} \circ π$ , where $π : G \to G / N$ is the canonical quotient map;
$V$ is irreducible as a representation of $G$ if and only if it is irreducible as a representation of $G / N$ .

Facts & Assumptions

Given: A group $G$ , a normal subgroup $N ⊴ G$ , a field $k$ , and a representation $ρ : G \to GL (V)$ with $N \subseteq \ker ρ$ .

[F1]

The kernel of a group homomorphism is the preimage of the identity (The kernel and image of a group homomorphism).

[F2]

The canonical map $π : G \to G / N$ , $π (g) = g N$ , is a surjective group homomorphism (The canonical projection $π : G \to G / N$ , $π (g) = g N$ , is a surjective group homomorphism).

[F3]

The cosets form the group $G / N$ under $(g N) (h N) = g h N$ , with identity $N$ and inverse $g^{- 1} N$ (For $N ⊴ G$ , the cosets form a group with identity $N$ and inverse $(g N)^{- 1} = g^{- 1} N$ ).

Proof

technique · direct

1.1

If $g N = h N$ , then $g^{- 1} h \in N$ by the coset laws of [F3], so $ρ (g^{- 1} h) = e$ by [F1] and the hypothesis $N \subseteq \ker ρ$ ; hence $ρ (h) = ρ (g)$ . Therefore $\overline{ρ} (g N) : = ρ (g)$ is independent of the chosen coset representative.

F1F3given

2.1

By [F3], $(g N) (h N) = g h N$ ; applying $ρ$ gives $\overline{ρ} (g N) \overline{ρ} (h N) = ρ (g) ρ (h) = ρ (g h) = \overline{ρ} (g h N) = \overline{ρ} ((g N) (h N))$ , so $\overline{ρ}$ is a group homomorphism into $GL (V)$ , with $ρ = \overline{ρ} \circ π$ by [F2]. This proves claim 1.

F2F3step 1.1given

3.1

A subspace $U \subseteq V$ is $ρ$ -stable exactly when it is $\overline{ρ}$ -stable, because $π$ of [F2] is surjective and $ρ (g) = \overline{ρ} (g N)$ : the two stability conditions quantify over the same operators.

F2step 2.1given

4.1

A representation is irreducible exactly when its only stable subspaces are $0$ and $V$ . By step 3.1 the stable subspaces for $ρ$ and for $\overline{ρ}$ coincide, so the two irreducibility statements are equivalent, which is claim 2.

step 3.1algebra∎

Depends on

Used by

The normal subgroups of a finite group are exactly the intersections of kernels of irreducible complex characters Theorem

Dependency tree · two levels

11 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, Section 4.2 (standard reference, not scraped)