# Lagrange theorem for finite groups

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

Equal-size left cosets of a subgroup H partition the finite group G, so |G|=[G:H]*|H| and |H| divides |G|.

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



## Mathematical statement

The complete Cayley table must certify H as a subgroup. Every left coset gH is equal to or disjoint from every other and all cosets cover G. Left translation h→g*h bijects H with gH, so every coset has size |H|, giving |G|=[G:H]*|H|. The converse family here only shows that a specified divisor-sized subset need not be a subgroup; its witness neither proves nor refutes existence of some subgroup of that order.





## Representative formula (exact TeX)

c(G)=i(G,H)*c(H)





## Assumptions


- The complete finite Cayley table, identity and candidate subset are explicit.
- Positive families check identity/product/inverse closure and enumerate every coset.





## Symbol scope


- c — finite cardinality
- G — finite group
- i — subgroup index
- H — certified subgroup





## Intuition, definition, and misconceptions

Left translation is a bijection from H to gH.

Cosets are equal or disjoint and cover G.

The divisibility conclusion supplies no unconditional converse.

A subset size dividing the group order does not certify that subset as a subgroup; a failed candidate is not a certificate that no subgroup of that order exists.





## Adopted derivation guide

[Lagrange counting from all left cosets](https://ocme.evemisslab.com/en/derivations/derivation-mko-lagrange-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


- [Finite group](https://ocme.evemisslab.com/en/mko/mko-finite-group/) — Use the complete Cayley table.
- [Subgroups and group homomorphisms](https://ocme.evemisslab.com/en/mko/mko-subgroup-homomorphism/) — Subgroup closure is required.
- [Natural-number divisibility](https://ocme.evemisslab.com/en/mko/mko-natural-divisibility/) — Compare group and subgroup orders.
- [Cardinality comparison of sets](https://ocme.evemisslab.com/en/mko/mko-cardinality-comparison/) — Count a finite equal-size partition.
- [Quotient set](https://ocme.evemisslab.com/en/mko/mko-quotient-set/) — Left cosets form a quotient partition.





## Verification boundary

Only complete finite group tables are handled; sampled products do not certify subgroup closure.

not_formalized

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





## Next obligation

Add the finite-group Cauchy theorem.





## 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/lagrange-theorem-finite-groups.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-lagrange-theorem-finite-groups/index.md)