PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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.

If $H \leq P$ are finite $p$ -groups, then $Φ (H) \leq Φ (P)$

Statement

If $H \leq P$ are finite $p$ -groups, then $Φ (H) \leq Φ (P)$ .

Facts & Assumptions

Given: Finite $p$ -groups $H \leq P$ .

[L1]

For every finite $p$ -group $P$ , $Φ (P) = P^{'} P^{p}$ ( $Φ (P) = P^{'} P^{p}$ for a finite $p$ -group).

[L2]

For every subgroup $H \leq P$ , one has $H^{'} \leq P^{'}$ (Homomorphisms respect commutator subgroups and derived series).

Proof

technique · direct

1.1

By [L2], $H^{'} \leq P^{'}$ . Every $p$ th power of an element of $H$ is also a $p$ th power of an element of $P$ , so $H^{p} \leq P^{p}$ .

givenL1L2algebra

2.1

Multiplying the inclusions in step 1.1 gives $H^{'} H^{p} \leq P^{'} P^{p}$ , and [L1] identifies these products with $Φ (H)$ and $Φ (P)$ .

step 1.1L1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

M. van Beek, Topics in Finite p-Groups, Lemma 3.6(i) (standard reference, not scraped)

If H≤PH≤P are finite pp-groups, then Φ(H)≤Φ(P)Φ(H)≤Φ(P)