LemmaStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 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.

Elementary groups are supersolvable

Statement

Every finite $p$ -elementary group is supersolvable.

Facts & Assumptions

[F1]

The cited prerequisite is Every nontrivial finite $p$ -group has nontrivial center, in fact $p$ divides $∣ Z (P) ∣$ .

Proof

Given: $G = C \times P$ with $C$ cyclic of $p^{'}$ -order and $P$ a finite $p$ -group.

1.1

A cyclic group has a normal series with prime-order factors. The case $P = 1$ is immediate. If $P \neq 1$ , its centre contains a subgroup $Z$ of order $p$ , and $P / Z$ has smaller order; by the induction hypothesis it has a normal prime-factor series, whose inverse images preceded by $1 ◃ Z$ give one for $P$ .

F1givenbaseih

2.1

Concatenate the series for $C$ and the series $C \times P_{i}$ from the series of $P$ . Each term is normal in $C \times P$ and every factor has prime order, including the cases $C = 1$ or $P = 1$ . ∎

step 1.1discharge-induction

Depends on

Used by

Characters of elementary groups are induced from linear characters Lemma

Dependency tree · two levels

17 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

Wen-Wei Li, Yanqi Lake Lectures on Algebra I, Remark 14.1.3 (standard reference, not scraped)