# A complete six-element action on four points

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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-group-action-01 · ocme-new142-reader-adoption-20261021



## Goal and scope

The explicit 24-entry table is a valid group action, but it is neither free nor transitive.

Complete Cayley/action tables and universal formulas; permutation rows alone do not qualify an action.





## 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). Points X={0,1,2,3}; (i,j) acts on x<3 by i+(-1)^j*x mod3 and fixes 3.


- G — six-element group carrier
- r,s — labels with r^3=e,s^2=e,sr=r^2s
- X — four points





## 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 — Use one complete finite Cayley table, complete point carrier and total action table. Check e*x=x and (g*h)*x=g*(h*x) for all relevant indices. Each group element then induces a permutation, but validity alone does not certify faithful, free or transitive action. Sampled pairs are insufficient. · mko-finite-group-action





## 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

X=\{0,1,2,3\}

After

\begin{array}{c|cccc}\cdot&0&1&2&3\\\hline e&0&1&2&3\\r&1&2&0&3\\r^2&2&0&1&3\\s&0&2&1&3\\rs&1&0&2&3\\r^2s&2&1&0&3\\\end{array}

Conditions: none

Rule: calc · Uses: G1, S02

Leaf basis: calc

Expand all 24 action values. For x<3, g*(h*x)=i+(-1)^j*k+(-1)^(j+l)*x=(gh)*x; both sides fix 3, and e fixes every point.



S04 · impliesBefore

g^{-1}(g\cdot x)=(g^{-1}g)\cdot x=e\cdot x=x

After

x\mapsto g\cdot x\text{ has inverse }x\mapsto g^{-1}\cdot x

Conditions: none

Rule: calc · Uses: G1, S02, S03

Leaf basis: calc

The established action axioms and group inverse supply an inverse function for each row, so every g induces a permutation.



S05 · impliesBefore

s\ne e,\quad s\cdot0=0

After

\text{the action is not free}

Conditions: none

Rule: calc · Uses: G1, S01, S03

Leaf basis: calc

Nonidentity s fixes point 0, directly refuting freeness.



S06 · impliesBefore

g\cdot0\in\{0,1,2\}\quad\forall g\in G

After

3\notin G\cdot0\Rightarrow\text{not transitive}

Conditions: none

Rule: calc · Uses: G1, S03

Leaf basis: calc

The complete 0-column contains only 0,1,2; no g sends 0 to 3, so the action is not transitive.



S07 · impliesBefore

K=\{e,u\},\ u^2=e,\quad Y=\{0,1\}

After

\begin{array}{c|cc}*&e&u\\\hline e&e&u\\u&u&e\end{array}\quad\begin{array}{c|cc}\cdot&0&1\\\hline e&0&1\\u&0&1\end{array}

Conditions: none

Rule: calc · Uses: G1, S03, S04

Leaf basis: calc

A complete C2 table and trivial action are shown: every element acts identically, so compatibility holds, but nonidentity u also acts identically and the action is not faithful.



S08 · impliesBefore

e\cdot x=x,\quad(gh)\cdot x=g\cdot(h\cdot x)\text{ for all indices}

After

G\curvearrowright X\text{ is qualified; free/faithful/transitive require separate certificates}

Conditions: none

Rule: mko_definition · Uses: G1, S03, S04, S05, S06, S07

Leaf basis: mko_definition

Identity and compatibility are fully checked in the same model; extra properties cannot be added automatically.







## Substeps and lemmas

No substeps in this guide.





## Reverse audits


- S08 · premise_audit · G1, S03, S04, S05, S06, S07 — S05,S06,S07 illustrate three distinct extra obligations rather than folding them into the action definition.
- S07 · premise_audit · G1, S03, S04 — The small group's laws descend from addition modulo 2, and both sides of trivial-action compatibility are x.
- S06 · premise_audit · G1, S03 — 3 is part of the point carrier and is fixed by all g; it cannot be removed to alter the conclusion.
- S05 · premise_audit · G1, S01, S03 — A valid action need not be free, so this does not contradict S03.
- S04 · premise_audit · G1, S02, S03 — Being permutations follows from the axioms; visually permutation-like rows do not prove those axioms.
- S03 · premise_audit · G1, S02 — Every group-point pair has one value; the formula covers all 144 triples, not merely permutation rows.
- 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.


- Permutation rows always give an action. → reject — On two points assign C3 rows e=id,r=swap,r2=id; r2*r=e but id composed with swap is not id, so compatibility fails.
- Every valid action is free. → reject — Here s is not e yet fixes 0.
- Every valid action is transitive or faithful. → reject — The two-point trivial C2 action in S07 is neither transitive nor faithful.





## Result and limitations

The explicit 24-entry table is a valid group action, but it is neither free nor transitive.







## Verification and source binding

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


- source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
- adopted source SHA-256: 66876f6517e8e0f296b1b8edfb9302c71f46b693da283ac405ce212032fd0684
- original source SHA-256: 66876f6517e8e0f296b1b8edfb9302c71f46b693da283ac405ce212032fd0684
- Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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