LemmaStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedaudited 2026-09-06

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Monomiality lifts along a quotient

Statement

Let $N ◃ G$ . If every irreducible character of $G / N$ is monomial, then every irreducible character of $G$ with $N$ in its kernel is monomial.

Facts & Assumptions

[F1]

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

Proof

Given: $χ$ is irreducible with $N \subseteq \ker χ$ .

1.1

Factor $χ$ through an irreducible character $\overset{ˉ}{χ}$ of $G / N$ . By hypothesis choose $\overset{ˉ}{H} \leq G / N$ and a linear $\overset{ˉ}{λ}$ with $\overset{ˉ}{χ} = {Ind}_{\overset{ˉ}{H}}^{G / N} \overset{ˉ}{λ}$ .

F1givenconstruct

2.1

Let $H$ be the inverse image of $\overset{ˉ}{H}$ and inflate $\overset{ˉ}{λ}$ to a linear $λ$ of $H$ . Compatibility of induction with quotient inflation gives $χ = {Ind}_{H}^{G} λ$ . ∎

step 1.1discharge-construct

Depends on

Used by

Finite supersolvable groups are monomial Theorem

Dependency tree · two levels

14 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

Tammo tom Dieck, Representation Theory, Lemma 4.3.4 (standard reference, not scraped)