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.

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A nonabelian supersolvable group has a noncentral normal abelian subgroup

Statement

Every nonabelian finite supersolvable group has an abelian normal subgroup that is not central.

Facts & Assumptions

[F1]

The cited prerequisite is Supersolvable groups and monomial characters.

Proof

Given: $1 = G_{0} ◃ \dots ◃ G_{r} = G$ is a supersolvable series.

1.1

Let $i$ be maximal with $G_{i}$ abelian. If $i = r$ , $G$ is abelian, so $i < r$ . By maximality there is $g \in G$ not commuting with some $a \in G_{i}$ .

F1given

2.1

Since $G_{i} ◃ G$ , it is an abelian normal subgroup, while the chosen $g, a$ show that it is not central. ∎

step 1.1

Depends on

Used by

Finite supersolvable groups are monomial Theorem

Dependency tree · two levels

8 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 4.3.3 (standard reference, not scraped)