# Exact-order witnesses for the prime divisors

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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-cauchy-theorem-finite-groups-01 · ocme-new142-reader-adoption-20261021



## Goal and scope

In this six-element group, prime divisors 2,3 have exact-order witnesses s,r; there is no element of order 6.

Complete element-order enumeration in a specified finite model, not a general Cauchy proof.





## 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
- p — prime divisor
- \operatorname{ord}(g) — least positive identity-return exponent





## 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 — A complete Cayley table must certify G as a finite group, and p must be prime and divide |G|. There exists g with g^p=e while g^k≠e for every1≤k<p, hence ord(g)=p. Complete finite enumeration here supplies witnesses for a specified model, not a replacement for the general theorem proof. Do not extend the prime-divisor conclusion to every composite divisor d of |G|. · mko-cauchy-theorem-finite-groups





## 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|=6

After

6=2\cdot3,\quad2,3\text{ prime},\quad2\mid6,3\mid6

Conditions: none

Rule: calc · Uses: G1, S01, S02

Leaf basis: calc

The qualified group has six elements and prime divisors exactly 2 and 3, both meeting the theorem premise.



S04 · impliesBefore

p=2,\quad g=s

After

s^1=s\ne e,\quad s^2=e\Rightarrow\operatorname{ord}(s)=2

Conditions: none

Rule: calc · Uses: G1, S01, S03

Leaf basis: calc

Check the only smaller positive exponent 1 and return at exponent 2, proving exact order 2 rather than merely g^2=e.



S05 · impliesBefore

p=3,\quad g=r

After

r^1=r\ne e,\quad r^2\ne e,\quad r^3=e\Rightarrow\operatorname{ord}(r)=3

Conditions: none

Rule: calc · Uses: G1, S01, S03

Leaf basis: calc

Both smaller positive powers are nonidentity and the third returns, proving exact order 3.



S06 · impliesBefore

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

After

\operatorname{ord}(g)=1,3,3,2,2,2\text{ respectively}

Conditions: none

Rule: calc · Uses: G1, S01, S02, S04, S05

Leaf basis: calc

Powers of r^2 are r^2,r,e; each reflection is nonidentity at power 1 and identity at power 2, completing all six orders.



S07 · impliesBefore

6\mid|G|,\quad6\notin\{1,2,3\}

After

\nexists g\in G:\operatorname{ord}(g)=6

Conditions: none

Rule: calc · Uses: G1, S03, S06

Leaf basis: calc

The complete order table excludes an element of composite order 6, showing that the prime conclusion does not extend to every divisor.



S08 · impliesBefore

p\in\{2,3\}\text{ prime divisor of }6

After

\exists g\in G:\ g^p=e\text{ and }g^k\ne e\ (1\le k<p)

Conditions: none

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

Leaf basis: mko_definition

Each prime has a full exact-order witness, certifying Cauchy instances in this model.







## Substeps and lemmas

No substeps in this guide.





## Reverse audits


- S08 · premise_audit · G1, S03, S04, S05, S06, S07 — The theorem for all finite groups still requires a general proof, which this finite table does not replace.
- S07 · premise_audit · G1, S03, S06 — Nonexistence follows from exhaustive enumeration, not a supposed Cauchy converse.
- S06 · premise_audit · G1, S01, S02, S04, S05 — Every element appears once, not just the generators r,s.
- S05 · premise_audit · G1, S01, S03 — r^3=e alone is insufficient: e also satisfies it but has order 1.
- S04 · premise_audit · G1, S01, S03 — s and e are distinct table labels, excluding order 1.
- S03 · premise_audit · G1, S01, S02 — 6 divides 6 but is not prime, so it cannot be the theorem's p.
- 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.


- e^3=e makes e order 3. → reject — e^1=e, so its least positive return is 1.
- 6 divides the group order, so an element has order 6. → reject — The full six-entry order table contains no 6.
- One six-element example proves general Cauchy. → reject — It certifies the specified model, not the quantification over all groups.





## Result and limitations

In this six-element group, prime divisors 2,3 have exact-order witnesses s,r; there is no element of order 6.







## Verification and source binding

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


- source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
- adopted source SHA-256: 9540176dd0ce1cca78a7c16996e5fc3ace68a2130c439dd5bdcc4779f164108a
- original source SHA-256: 9540176dd0ce1cca78a7c16996e5fc3ace68a2130c439dd5bdcc4779f164108a
- Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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