Full conjugation action and three classes

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

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)

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

\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}

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

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

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

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

x=s,rs,r^2s

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

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

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

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

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

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

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

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 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.

• 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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