A quotient table checked on every representative pair

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

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

\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}

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

((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)

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

\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}

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

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

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

\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}

Q=\{A,B\}

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

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

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

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

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

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

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

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 table

Bind every product, inverse and coset to this complete Cayley table.

Complete enumeration

Every 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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