# Cauchy theorem for finite groups

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

If a prime p divides the order of a finite group G, G contains an element of exact order p.

mko-cauchy-theorem-finite-groups · theorem · v0.10.0



## Mathematical statement

A complete Cayley table must certify G as a finite group, and p must be prime and divide |G|. There exists g with g^p=e while g^k≠e for every1≤k<p, hence ord(g)=p. Complete finite enumeration here supplies witnesses for a specified model, not a replacement for the general theorem proof. Do not extend the prime-divisor conclusion to every composite divisor d of |G|.





## Representative formula (exact TeX)

o(g)=p





## Assumptions


- Positive families use complete closed finite Cayley tables passing associativity, identity and inverse checks.
- Element order is the least positive identity-return exponent, and every smaller positive power is checked.





## Symbol scope


- o — element order
- g — finite-group element
- p — prime divisor of the group order





## Intuition, definition, and misconceptions

For every prime divisor there is at least one exact-order-p witness.

An order-p certificate checks the p-th power and every smaller positive power.

Exhausting a finite table certifies only that group; the general Cauchy theorem remains a theorem-level obligation.

g^p=e alone does not establish exact order p; a prime-divisor conclusion cannot be copied to arbitrary composite divisors.





## Adopted derivation guide

[Exact-order witnesses for the prime divisors](https://ocme.evemisslab.com/en/derivations/derivation-mko-cauchy-theorem-finite-groups-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Lagrange theorem for finite groups](https://ocme.evemisslab.com/en/mko/mko-lagrange-theorem-finite-groups/) — The generated cyclic-subgroup order divides the group order.
- [Prime natural number](https://ocme.evemisslab.com/en/mko/mko-prime-number/) — p must qualify as prime.
- [Finite group](https://ocme.evemisslab.com/en/mko/mko-finite-group/) — The group operation is complete.
- [Finite group action](https://ocme.evemisslab.com/en/mko/mko-finite-group-action/) — General proofs may use actions and orbit counting.





## Verification boundary

Finite models do not replace the general proof, and not every composite divisor need be an element order.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add Sylow theorems and finite-group structure consequences.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/cauchy-theorem-finite-groups.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-cauchy-theorem-finite-groups/index.md)