# Finite quotient group

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

Cosets of a normal subgroup N form a finite quotient group, with (gN)(hN)=(g*h)N independent of representatives.

mko-finite-quotient-group · definition · v0.10.0



## Mathematical statement

First certify N as a normal subgroup of the complete group table. All left cosets form a disjoint complete partition; quotient elements are coset sets, not representatives. Record multiplication by coset indices and verify every replacement of either representative lands in the same product coset. N is the identity coset and g^(-1)N is the inverse. A nonnormal subgroup can have representative-dependent products, blocking this quotient multiplication. Quotients are not automatically commutative, and no first-isomorphism claim is added.





## Representative formula (exact TeX)

Q(G,N)=C





## Assumptions


- Families use complete finite Cayley tables, an identity and an explicit nonempty subset.
- Coset partitions and every representative pair are exhaustively enumerated.





## Symbol scope


- Q — quotient-group operator
- G — finite group
- N — normal subgroup
- C — coset carrier





## Intuition, definition, and misconceptions

Build the complete coset partition first.

Quotient multiplication must survive every representative substitution.

A nonnormal witness directly refutes well-definedness.

A representative is not the quotient element itself, and cosets cannot simply be multiplied without normality.





## Adopted derivation guide

[A quotient table checked on every representative pair](https://ocme.evemisslab.com/en/derivations/derivation-mko-finite-quotient-group-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Finite normal subgroup](https://ocme.evemisslab.com/en/mko/mko-finite-normal-subgroup/) — Normality is needed for well-defined products.
- [Quotient set](https://ocme.evemisslab.com/en/mko/mko-quotient-set/) — Cosets are equivalence classes.
- [Finite group](https://ocme.evemisslab.com/en/mko/mko-finite-group/) — Use the original multiplication and inverses.





## Verification boundary

Only the finite quotient construction is made; commutativity or isomorphism with an image is not automatic.

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 first isomorphism theorem for finite groups.





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