CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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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.

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No group of order $p^{2} q$ for distinct primes is simple

Statement

No group of order $p^{2} q$ for distinct primes is simple. See Every group of order $p^{2} q$ for distinct primes has a normal Sylow subgroup.

Facts & Assumptions

Given: The hypotheses and objects in the Statement.

[L1]

Every group of order $p^{2} q$ for distinct primes has a normal Sylow subgroup. (Every group of order $p^{2} q$ for distinct primes has a normal Sylow subgroup).

[L2]

A group $G$ is simple if $G \neq {1}$ and its only normal subgroups are ${1}$ and $G$ , where normality is as in def-normal-subgroup. (Simple groups).

Proof

technique · direct

1.1

The normal Sylow subgroup supplied by the theorem has prime-power order strictly between one and $p^{2} q$ , so it is a nontrivial proper normal subgroup.

L1L2givenalgebra

2.1

Both cases of step 1.1 are genuinely nontrivial and proper: a normal Sylow $p$ -subgroup has order $p^{2}$ with $1 < p^{2} < p^{2} q$ because $q > 1$ , and a normal Sylow $q$ -subgroup has order $q$ with $1 < q < p^{2} q$ because $p^{2} > 1$ . Either one is therefore a normal subgroup other than ${1}$ and $G$ , which is what [L2] requires for $G$ to fail simplicity. This proves the stated claim.

step 1.1L2givenalgebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 12 results over 5 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)

No group of order p2qp2q for distinct primes is simple