LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 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.

Every subgroup of a finite $p$ -group has order a power of $p$

Statement

If $P$ is a finite $p$ -group and $H\le P$ , then $H$ is finite and

$|H|=p^k$

for some $k\in\mathbb N$ . If $|P|=p^n$ , then $k\le n$ .

Facts & Assumptions

Given: A finite $p$ -group $P$ with $|P|=p^n$ and a subgroup $H\le P$ .

[L1]

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

[L2]

Lagrange gives $|P|=[P:H]|H|$ (Lagrange's theorem: $|G|=[G:H]|H|$ for every subgroup $H$ of a finite group $G$ ).

[L3]

Positive integers have unique prime factorisations (For $n \ge 1$ and any injective list $p : r \to \mathbb{Z}$ of primes containing every prime divisor of $n$ , one has $n = \prod_{i<r} p_i^{\,v_{p_i}(n)}$ ; the exponents are determined by $n$ , and $v_q(n) = 0$ for every prime $q$ outside the list).

Proof

technique · direct

1.1

By [L2], the finite set $H$ has positive order dividing $p^n$ .

L1L2

2.1

By uniqueness in [L3], no prime other than $p$ can divide $|H|$ , so $|H|=p^k$ for some natural $k\le n$ .

step 1.1L3

3.1

This includes the trivial subgroup, whose order is $1=p^0$ , and proves that every subgroup of $P$ is a finite $p$ -group.

step 2.1algebra∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 109 results over 25 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)

Every subgroup of a finite ppp-group has order a power of ppp