# Certifying the center from all commutation rows

[← Guide index](https://ocme.evemisslab.com/en/derivations/) · [Original MKO](https://ocme.evemisslab.com/en/mko/mko-finite-group-center/)

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-center-01 · ocme-new142-reader-adoption-20261021



## Goal and scope

Z(G)={e}, directly a normal subgroup; G is nonabelian and r is not central despite commuting with itself.

A complete six-by-six commutation table, closure and all conjugations; no characteristic or class-equation claim.





## 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
- T,F — truth values of full commutation tests
- Z(G) — elements whose rows are all true





## 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 — From a complete qualified Cayley table, build each element z's commutation row against every g. The center Z(G) consists exactly of all-true rows. Directly check identity, product and inverse closure, then every conjugation to certify normality. z is central exactly when its full conjugacy class is a singleton, and Z(G)=G exactly when G is abelian. The center requires commutation with every group element, whereas a single-element centralizer imposes only that one commutation condition; they must not be identified in general, although they may coincide in particular groups. Characteristic status and the class equation are not asserted in this slice. · mko-finite-group-center





## 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

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

After

\begin{array}{c|cccccc}gx=xg&e&r&r^2&s&rs&r^2s\\\hline e&T&T&T&T&T&T\\r&T&T&T&F&F&F\\r^2&T&T&T&F&F&F\\s&T&F&F&T&F&F\\rs&T&F&F&F&T&F\\r^2s&T&F&F&F&F&T\\\end{array}

Conditions: none

Rule: calc · Uses: G1, S01, S02

Leaf basis: calc

All 36 booleans record zg=gz, comparing each row with all six g values.



S04 · impliesBefore

Z(G)=\{z:\ zg=gz\ \forall g\in G\}

After

Z(G)=\{e\}

Conditions: none

Rule: mko_definition · Uses: G1, S03

Leaf basis: mko_definition

Only e's row is all true; each other row contains false, so the full center contains only the identity.



S05 · impliesBefore

Z=\{e\}

After

e\in Z,\quad ee=e\in Z,\quad e^{-1}=e\in Z\Rightarrow Z\le G

Conditions: none

Rule: calc · Uses: G1, S02, S04

Leaf basis: calc

For this singleton carrier, identity, the sole product and the sole inverse are directly checked.



S06 · impliesBefore

g=e,r,r^2,s,rs,r^2s,\quad z=e

After

geg^{-1}=e,e,e,e,e,e\text{ respectively}\Rightarrow Z\trianglelefteq G

Conditions: none

Rule: calc · Uses: G1, S02, S04, S05

Leaf basis: calc

All six conjugations preserve the sole central element, directly proving normality.



S07 · equivalentBefore

gxg^{-1}=x\ \forall g

After

gx=xg\ \forall g\iff x\in Z(G)\iff\operatorname{Cl}(x)=\{x\}

Conditions: none

Rule: calc · Uses: G1, S02, S03, S04

Leaf basis: calc

Multiplication by g and the reverse by g inverse prove singleton class iff centrality; only e qualifies here.



S08 · impliesBefore

rs\ne sr=r^2s

After

G\text{ not abelian};\quad Z(G)\ne G;\quad Z(G)=G\iff G\text{ abelian}

Conditions: none

Rule: mko_definition · Uses: G1, S01, S03, S04, S07

Leaf basis: mko_definition

An explicit noncommuting pair refutes abelianness; the general equivalence restates the same universal commutation condition.







## Substeps and lemmas

No substeps in this guide.





## Reverse audits


- S08 · premise_audit · G1, S01, S03, S04, S07 — The elements commuting with fixed r form the larger set {e,r,r2}; that single-element centralizer is not Z(G).
- S07 · inverse_equivalence · G1, S02, S03, S04 — No class-size formula or class equation is needed, only universal commutation.
- S06 · premise_audit · G1, S02, S04, S05 — Normality and subgroup status are separate checks; the preceding step cannot replace this one.
- S05 · premise_audit · G1, S02, S04 — Subgroup status is not skipped because of the name center; all finite obligations are met.
- S04 · premise_audit · G1, S03 — True entries against e,r,r2 are insufficient for r; the false entry against s refutes centrality.
- S03 · premise_audit · G1, S01, S02 — T comes from equality of two products in the same table, not a test against just one g.
- 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.


- r commutes with r, hence r is central. → reject — r does not commute with s; centrality requires every g.
- Z(G)={e,r,r2}. → reject — That is the set commuting with r; r,r2 do not commute with s.
- A finite group is abelian. → reject — This complete six-element table has rs distinct from sr.





## Result and limitations

Z(G)={e}, directly a normal subgroup; G is nonabelian and r is not central despite commuting with itself.







## Verification and source binding

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


- source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
- adopted source SHA-256: a09d76035cc0fb9c7b94984a4710a247026139a690e8d946c8fb33de53d58b04
- original source SHA-256: b8003d263c41d04f2bf8df3d9db33a054c738be3d16e50080408676978cb6c9f
- Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

[Same-source full JSON (formulas, ASTs, controls and revision log)](https://ocme.evemisslab.com/data/derivations/coverage-20261021/guides/derivation-mko-finite-group-center-01.json)