CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Normal subgroup induction criterion

Statement

Let $G$ be finite, $N ⊴ G$ , and $θ \in Irr (N)$ . Then ${⟨ {Ind}_{N}^{G} θ, {Ind}_{N}^{G} θ ⟩}_{G} = [I_{G} (θ) : N] .$ In particular, ${Ind}_{N}^{G} θ$ is irreducible if and only if $I_{G} (θ) = N$ .

Facts & Assumptions

Given: The groups, modules, characters, and hypotheses in the statement. All representations here are finite-dimensional complex left representations.

[F1]

$I_{G} (θ)$ is the stabilizer of $θ$ for left conjugation and contains $N$ . (Inertia group and characters lying above a normal type).

[F2]

The induced function module is a vector-space direct sum of copies of its inducing module, one supported on each left coset. (A left transversal identifies ${Ind}_{H}^{G} W$ with a direct sum of $[G : H]$ copies of $W$ ).

[F3]

For finite groups and complex characters, $⟨ {Ind}_{H}^{G} α, β ⟩_{G} = ⟨ α, {Res}_{H}^{G} β ⟩_{H}$ . (Frobenius reciprocity for complex characters).

[F4]

The multiplicity of a simple constituent in a finite-dimensional complex representation is its character inner product with the representation character. (The multiplicity of an irreducible summand is a character inner product).

[F5]

A complex character of a finite group is irreducible if and only if its self-inner-product is one. (A complex character is irreducible if and only if its self-inner-product is $1$ ).

Proof

technique · direct

1.1

Let $S$ afford $θ$ and let $T$ meet the left cosets of $N$ . On the functions supported on $t N$ , evaluation satisfies $(n f) (t) = (t^{- 1} n t) f (t)$ . Therefore restriction of induction has character $\sum_{t \in T}^{t} θ$ , by taking traces on this finite direct sum.

F2givenalgebra

2.1

Applying the multiplicity formula to a simple module shows that two irreducible characters have inner product one when equal and zero otherwise. Conjugate characters are irreducible. Thus reciprocity gives $⟨ Ind θ, Ind θ ⟩_{G} = \sum_{t \in T} ⟨ θ,^{t} θ ⟩_{N}$ . Exactly the cosets $t N$ in $I_{G} (θ) / N$ contribute one, giving the asserted index.

F1F3F4step 1.1algebra

3.1

The induced character is an actual nonzero character. If it is irreducible its norm is one, hence $[I_{G} (θ) : N] = 1$ and $I_{G} (θ) = N$ . Conversely that equality makes its norm one, hence it is irreducible. This includes $N = G$ , when induction is identity, and $N = 1$ , when the norm is $∣ G ∣$ .

F5step 2.1algebra∎

Depends on

Used by

Ramification indices account for the inertia quotient Corollary
Clifford correspondence for A3 in S3 Example

Dependency tree · two levels

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Sources

Tammo tom Dieck, Representation Theory — Proposition 4.2.3 and equations (4.4)–(4.5) pp.54–55 (standard reference, not scraped)