Alphabeta MathHow 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.
No group of order for distinct primes is simple
Statement
No group of order for distinct primes is simple. See If are primes and , then has a normal subgroup of order .
Facts & Assumptions
Given: The hypotheses and objects in the Statement.
Let be primes. Every group of order has a normal subgroup of order . (If are primes and , then has a normal subgroup of order ).
A group is simple if and its only normal subgroups are and , where normality is as in def-normal-subgroup. (Simple groups).
Proof
We order the primes as .
The published order- lemma supplies a normal subgroup of order , which is nontrivial and proper. This proves the stated claim.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
14 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
- Keith Conrad, Consequences of the Sylow Theorems, Sections 1-5 (standard reference, not scraped)