DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-11

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.

A finite $p$ -group has order $p^n$ for a prime $p$ and some $n\in\mathbb N$

Definition

Let $p$ be a prime natural number (Prime and composite integers: $p$ is prime when $p > 1$ and its only positive divisors are $1$ and $p$ ). A finite $p$ -group is a finite group $P$ (Group and abelian group, The cardinality $\lvert A\rvert$ of a finite set) whose order has the form

$|P|=p^n$

for some $n\in\mathbb N$ , with natural exponentiation as in Exponentiation of natural numbers, $m^{n}$ , and its agreement with the integer power in $\mathbb{R}$ . The case $n=0$ permits the trivial group. A finite $p$ -group is nontrivial exactly when $n\ge 1$ .

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 80 results over 23 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

K. Conrad, Group Actions, Section 4 (standard reference, not scraped)

A finite ppp-group has order pnp^npn for a prime ppp and some n∈Nn\in\mathbb Nn∈N