TheoremStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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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.
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Finite supersolvable groups are monomial

Statement

Every finite supersolvable group is monomial.

Facts & Assumptions

[F1]

The cited prerequisite is A faithful irreducible is induced from a proper inertia subgroup.

Proof

Given: $G$ is finite supersolvable and $χ \in Irr (G)$ .

1.1

Induct on $∣ G ∣$ . The trivial group is the base case. If $G$ is abelian, $χ$ is linear. If $\ker χ \neq 1$ , then $G / \ker χ$ is supersolvable of smaller order and the quotient lemma makes $χ$ monomial.

F1givenbaseih

2.1

Otherwise $χ$ is faithful. A nonabelian $G$ has a noncentral normal abelian subgroup, so the proper-inertia proposition writes $χ = {Ind}_{H}^{G} θ$ with $H < G$ and $θ \in Irr (H)$ . Intersecting a supersolvable series with $H$ and deleting repeated terms makes $H$ supersolvable; by the induction hypothesis, $θ$ is induced from a linear character, and transitivity makes $χ$ so induced. ∎

step 1.1discharge-induction

Depends on

Used by

Characters of elementary groups are induced from linear characters Lemma

Dependency tree · two levels

20 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, Theorem 4.3.1 (standard reference, not scraped)