A complete six-element action on four points

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

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)

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

\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}

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

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

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

\text{the action is not free}

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

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

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

\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}

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

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

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 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.

• 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

Same-source full JSON (formulas, ASTs, controls and revision log)