DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-17

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.

$p$ -complements in finite groups

Definition

Let $G$ be finite and let $p$ be prime. A $p$ -complement is a subgroup $H \leq G$ whose order is prime to $p$ and whose index $[G : H]$ is a power of $p$ (The coset set $G / H$ and the index $[G : H]$ of a subgroup). Equivalently, when $∣ G ∣ = p^{a} m$ with $p ∤ m$ , a $p$ -complement has order $m$ . The definition asserts no existence.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Keith Conrad, The Sylow Theorems, Sections 1-2 (standard reference, not scraped)

pp-complements in finite groups