# A quotient table checked on every representative pair

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This is an owner-adopted natural-language teaching guide. Finite/arithmetic checks are not universal proofs; no Lean was run, and review scope is recorded per revision. Original source status labels are retained as historical data. Formulas retain exact TeX; no new semantic AST is invented.

derivation-mko-finite-quotient-group-01 · ocme-new142-reader-adoption-20261021



## Goal and scope

G/N has cosets A=N,B={s,rs,r2s} and well-defined products AA=A,AB=BA=B,BB=A.

Complete normality, partition and 36 representative-pair class indices, with a nonnormal failure witness.





## Givens and symbols


- calc


- G1 — G={e,r,r2,s,rs,r2s}, encoded respectively (0,0),(1,0),(2,0),(0,1),(1,1),(2,1), with (i,j)(k,l)=(i+(-1)^j*k mod3,j+l mod2).


- G — six-element group carrier
- r,s — labels with r^3=e,s^2=e,sr=r^2s
- A,B — whole quotient cosets





## Rules and sources


- calc · prerequisite — Assume exact arithmetic modulo 3 and 2, extensional finite-set equality, and definitions of groups/actions; this operation is not pre-assumed to be a group.
- mko_definition · cited — 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. · mko-finite-quotient-group





## Main derivation

S01 · impliesBefore

G=\{e,r,r^2,s,rs,r^2s\}

After

\begin{array}{c|cccccc}*&e&r&r^2&s&rs&r^2s\\\hline e&e&r&r^2&s&rs&r^2s\\r&r&r^2&e&rs&r^2s&s\\r^2&r^2&e&r&r^2s&s&rs\\s&s&r^2s&rs&e&r^2&r\\rs&rs&s&r^2s&r&e&r^2\\r^2s&r^2s&rs&s&r^2&r&e\\\end{array}

Conditions: none

Rule: calc · Uses: G1

Leaf basis: calc

Expand the pair formula for all 36 products; every output belongs to the same six-element carrier and no cell is missing.



S02 · impliesBefore

(i,j)(k,l)=(i+(-1)^jk,j+l)\quad(\bmod(3,2))

After

((i,j)(k,l))(m,n)=(i+(-1)^jk+(-1)^{j+l}m,j+l+n)=(i,j)((k,l)(m,n));\quad e=(0,0);\quad(i,j)^{-1}=(-(-1)^ji,j)

Conditions: none

Rule: calc · Uses: G1, S01

Leaf basis: calc

Both bracketings have identical coordinates, proving associativity for all 216 triples; (0,0) is a two-sided identity and the inverse formula multiplies back on both sides.



S03 · impliesBefore

N=\{e,r,r^2\}

After

\begin{array}{c|ccc}*&e&r&r^2\\\hline e&e&r&r^2\\r&r&r^2&e\\r^2&r^2&e&r\\\end{array}\qquad\begin{array}{c|ccc}gng^{-1}&e&r&r^2\\\hline e&e&r&r^2\\r&e&r&r^2\\r^2&e&r&r^2\\s&e&r^2&r\\rs&e&r^2&r\\r^2s&e&r^2&r\\\end{array}

Conditions: none

Rule: calc · Uses: G1, S01, S02

Leaf basis: calc

All nine products, inverses e,r^2,r and the complete 18-entry conjugation table qualify N as a normal subgroup first.



S04 · impliesBefore

A=N,\quad B=\{s,rs,r^2s\}

After

eN=rN=r^2N=A;\quad sN=(rs)N=(r^2s)N=B;\quad A\sqcup B=G

Conditions: none

Rule: calc · Uses: G1, S01, S03

Leaf basis: calc

All six left cosets deduplicate to two nonempty disjoint coset sets covering G.



S05 · impliesBefore

g,h=e,r,r^2,s,rs,r^2s

After

\begin{array}{c|cccccc}(gh)N&e&r&r^2&s&rs&r^2s\\\hline e&A&A&A&B&B&B\\r&A&A&A&B&B&B\\r^2&A&A&A&B&B&B\\s&B&B&B&A&A&A\\rs&B&B&B&A&A&A\\r^2s&B&B&B&A&A&A\\\end{array}

Conditions: none

Rule: calc · Uses: G1, S01, S04

Leaf basis: calc

All 36 entries record the coset of gh; each of the four 3-by-3 blocks has a single class.



S06 · impliesBefore

Q=\{A,B\}

After

\begin{array}{c|cc}*&A&B\\\hline A&A&B\\B&B&A\\\end{array}

Conditions: none

Rule: calc · Uses: G1, S04, S05

Leaf basis: calc

Compress the four constant blocks into the quotient multiplication table, with all four products well-defined.



S07 · impliesBefore

A=N,\quad B^{-1}=B

After

A\text{ identity},\quad A^{-1}=A,\quad B^{-1}=B

Conditions: none

Rule: calc · Uses: G1, S02, S06

Leaf basis: calc

The table directly checks identity and both inverses; these agree with the classes g inverse N.



S08 · impliesBefore

((gN)(hN))(kN)=(gh)kN

After

(gh)kN=g(hk)N=(gN)((hN)(kN))\Rightarrow G/N\text{ group}

Conditions: none

Rule: mko_definition · Uses: G1, S02, S05, S06, S07

Leaf basis: mko_definition

After well-definedness, associativity descends from G; closure, identity and inverses complete the quotient group.



S09 · impliesBefore

H=\{e,s\},\quad eH=sH=H

After

er=r\in rH=\{r,rs\};\quad sr=r^2s\in r^2H=\{r^2,r^2s\}

Conditions: none

Rule: calc · Uses: G1, S01, S02, S05, S08

Leaf basis: calc

For nonnormal H, representatives e and s of the same first coset multiplied by r land in different cosets, directly refuting well-defined multiplication.







## Substeps and lemmas

No substeps in this guide.





## Reverse audits


- S09 · premise_audit · G1, S01, S02, S05, S08 — H remains a subgroup by its four products and inverses; the failure is representative independence.
- S08 · premise_audit · G1, S02, S05, S06, S07 — Commutativity here comes from the actual two-element table, not a claim about every quotient.
- S07 · premise_audit · G1, S02, S06 — B^2=A, not B, so the reflection class is not the identity.
- S06 · premise_audit · G1, S04, S05 — The labels bind coset sets without adding parent-group elements.
- S05 · premise_audit · G1, S01, S04 — Thus any replacement of either representative within its coset preserves the product class, checking every pair rather than just e,s.
- S04 · premise_audit · G1, S01, S03 — Quotient elements are the full sets A,B, not representatives e,s.
- S03 · premise_audit · G1, S01, S02 — Qualification does not follow merely from writing G/N.
- S02 · premise_audit · G1, S01 — The sign depends only on j+l modulo 2 since (-1)^2=1; the full inverse list is e,r^2,r,s,rs,r^2s.
- S01 · premise_audit · G1 — The six labels bind the six pairs; r^2 is not an additional seventh element.





## Error controls and negative controls

Same group tableBind every product, inverse and coset to this complete Cayley table.

Complete enumerationEvery universal finite property needs the full carrier or a universal formula, not a proper sample.


- A is just the element e. → reject — A={e,r,r^2} is a three-element coset.
- Nonnormal H permits a quotient by choosing e,r representatives. → reject — e and s share a coset while er and sr have different product cosets.
- Every quotient group is commutative. → reject — For N={e}, the quotient retains sr=r^2s distinct from rs.





## Result and limitations

G/N has cosets A=N,B={s,rs,r2s} and well-defined products AA=A,AB=BA=B,BB=A.







## Verification and source binding

1 declared checks for this guide; only the listed expressions are covered.


- source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
- adopted source SHA-256: 9926dfd65633dc37bc3085815d1b8976ae602dc6244141c8032b0308293af92a
- original source SHA-256: 9926dfd65633dc37bc3085815d1b8976ae602dc6244141c8032b0308293af92a
- Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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