# Full conjugation action and three classes

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



## Goal and scope

The conjugacy classes are {e},{r,r2},{s,rs,r2s}; r and s are not conjugate.

Derive complete conjugation, orbits and partition from Cayley data without centralizers or a class-size formula.





## 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
- \operatorname{Cl}(x) — conjugacy class of element x





## Rules and sources

Some rule statements have reader clarifications in this revision: they are paraphrases, not verbatim quotations. Referenced original source bytes and hashes are unchanged.


- 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 — Derive the full conjugation table from the complete group table and inverses, checking identity and action composition. The class of x contains every g*x*g^(-1), and all classes are equal or disjoint and cover G. A singleton class means x commutes with every group element; an explicit y outside the class proves nonconjugacy. Conjugacy classes, cosets and conjugate subgroups are different constructions and must not be conflated, even when particular underlying sets coincide. No centralizer, class-size formula or additional orbit-stabilizer consequence is used here. · mko-finite-group-conjugacy-class





## 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

g,x\in G,\quad c_g(x)=gxg^{-1}

After

\begin{array}{c|cccccc}gxg^{-1}&e&r&r^2&s&rs&r^2s\\\hline e&e&r&r^2&s&rs&r^2s\\r&e&r&r^2&r^2s&s&rs\\r^2&e&r&r^2&rs&r^2s&s\\s&e&r^2&r&s&r^2s&rs\\rs&e&r^2&r&r^2s&rs&s\\r^2s&e&r^2&r&rs&s&r^2s\\\end{array}

Conditions: none

Rule: calc · Uses: G1, S01, S02

Leaf basis: calc

Compute every one of the 36 conjugation entries from the parent table and all inverses; the point carrier is G itself.



S04 · impliesBefore

c_e(x)=exe^{-1}=x

After

c_{gh}(x)=(gh)x(h^{-1}g^{-1})=g(hxh^{-1})g^{-1}=c_g(c_h(x))

Conditions: none

Rule: calc · Uses: G1, S02, S03

Leaf basis: calc

Group laws prove identity and composition for every index, qualifying the conjugation action.



S05 · impliesBefore

\operatorname{Cl}(x)=\{c_g(x):g\in G\}

After

\operatorname{Cl}(e)=\{e\},\quad\operatorname{Cl}(r)=\operatorname{Cl}(r^2)=\{r,r^2\}

Conditions: none

Rule: calc · Uses: G1, S03, S04

Leaf basis: calc

Deduplicate the complete columns for e,r,r2 to obtain the identity and rotation classes.



S06 · impliesBefore

x=s,rs,r^2s

After

\operatorname{Cl}(s)=\operatorname{Cl}(rs)=\operatorname{Cl}(r^2s)=\{s,rs,r^2s\}

Conditions: none

Rule: calc · Uses: G1, S03, S04, S05

Leaf basis: calc

The remaining three full columns each traverse the reflection set, completing all six elements' classes.



S07 · impliesBefore

C_0=\{e\},C_1=\{r,r^2\},C_2=\{s,rs,r^2s\}

After

C_i\cap C_j=\varnothing\ (i\ne j),\quad C_0\cup C_1\cup C_2=G

Conditions: none

Rule: mko_definition · Uses: G1, S05, S06

Leaf basis: mko_definition

The three distinct classes directly form a disjoint cover, agreeing with the equal-or-disjoint orbit property.



S08 · equivalentBefore

\operatorname{Cl}(x)=\{x\}

After

gxg^{-1}=x\ \forall g\iff gx=xg\ \forall g

Conditions: none

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

Leaf basis: calc

Right multiplication by g and the reverse by g inverse show singleton exactly when x commutes with every g; here only e has a singleton class.



S09 · impliesBefore

s\notin\operatorname{Cl}(r)=\{r,r^2\}

After

r\text{ and }s\text{ nonconjugate}

Conditions: none

Rule: calc · Uses: G1, S03, S05, S07

Leaf basis: calc

The full r-column excludes s, exhausting all g and certifying nonconjugacy.







## Substeps and lemmas

No substeps in this guide.





## Reverse audits


- S09 · premise_audit · G1, S03, S05, S07 — Failure for one g would not suffice; the complete column is used.
- S08 · inverse_equivalence · G1, S02, S05, S06, S07 — This uses only conjugation, not a class-size formula or later theorem.
- S07 · premise_audit · G1, S05, S06 — This is a class partition without a centralizer count.
- S06 · premise_audit · G1, S03, S04, S05 — Several elements share one class; deduplication leaves one reflection class.
- S05 · premise_audit · G1, S03, S04 — The calculation includes all reflections producing r2, not merely r conjugating itself.
- S04 · premise_audit · G1, S02, S03 — The inverse of gh is h inverse g inverse in reverse order; the wrong order would invalidate compatibility.
- S03 · premise_audit · G1, S01, S02 — Rows index g and columns x, not conjugate subgroups gHg inverse.
- 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.


- The class of r is only {r}. → reject — s*r*s inverse=r2 also belongs to the class.
- r and s are conjugate because both lie in G. → reject — The full class of r contains only r,r2, not s.
- Every conjugacy class is a subgroup. → reject — {r,r2} omits the identity e.





## Result and limitations

The conjugacy classes are {e},{r,r2},{s,rs,r2s}; r and s are not conjugate.







## Verification and source binding

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


- source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
- adopted source SHA-256: 18e9a1b4f67b6f1182f4512c4416da956b0d05cdac22fa05cd0d5f38ac074087
- original source SHA-256: a76c966c381c514457e23c2308b223bd35aee8d53f4c9b6b577beb4dce19d883
- Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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