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.

Subgroups of elementary and hyperelementary groups

Statement

Every subgroup of a finite $p$ -elementary group is $p$ -elementary, and every subgroup of a finite $p$ -hyperelementary group is $p$ -hyperelementary.

Facts & Assumptions

[F1]

The cited prerequisite is $p$ -elementary and $p$ -hyperelementary finite groups.

Proof

Given: $G = C ⋊ P$ is $p$ -hyperelementary and $H \leq G$ ; in the elementary case the action is trivial.

1.1

Put $C_{H} = H \cap C$ . It is cyclic, normal in $H$ , and $H / C_{H}$ embeds in $G / C ≅ P$ ; hence it is a $p$ -group. A Sylow $p$ -subgroup $P_{H}$ of $H$ maps isomorphically onto $H / C_{H}$ , because its image has the full $p$ -power order and $C_{H}$ has order prime to $p$ .

F1given

2.1

Thus $H = C_{H} ⋊ P_{H}$ . If $G = C \times P$ , its Sylow $p$ -subgroup is unique, so $P_{H} = H \cap P$ and it commutes with $C_{H}$ ; consequently $H = C_{H} \times P_{H}$ . ∎

step 1.1

Depends on

Used by

Hyperelementary permutation subring reduction Lemma

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

Wen-Wei Li, Yanqi Lake Lectures on Algebra I, Lemma 14.1.2 (standard reference, not scraped)