DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + claude-sonnet-5)audited 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.

Sylow $p$ -subgroups of a finite group

Definition

Let $G$ be a finite group, let $p$ be prime, and write $∣ G ∣ = p^{a} m$ with $a \in N$ and $p ∤ m$ . A subgroup $P \leq G$ is a Sylow $p$ -subgroup when $∣ P ∣ = p^{a}$ . Equivalently, its order is the largest power of $p$ dividing $∣ G ∣$ . This is a property of a subgroup and does not presume that such a subgroup exists; existence is proved in Sylow I: every finite group has a Sylow $p$ -subgroup.

Depends on

Used by

p-complements in finite groups Definition
The number nₚ(G) of Sylow p-subgroups Definition
Sylow p-subgroups of Aut((ℤ/p)²): nₚ=p+1 Example
Distinct normal Sylow subgroups centralize one another Lemma
Sylow I: every finite group has a Sylow p-subgroup Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 68 results over 21 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Keith Conrad, Consequences of the Sylow Theorems, Sections 1-5 (standard reference, not scraped)

Sylow pp-subgroups of a finite group