# Lagrange counting from all left cosets

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

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



## Goal and scope

H={e,s} has three distinct left cosets of size 2, giving 6=[G:H]|H|=3*2.

A full six-element table and full coset partition; a failed subset does not prove absence of subgroups of that order.





## 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
- H — specified order-2 subgroup
- [G:H] — number of distinct left cosets





## 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 — The complete Cayley table must certify H as a subgroup. Every left coset gH is equal to or disjoint from every other and all cosets cover G. Left translation h→g*h bijects H with gH, so every coset has size |H|, giving |G|=[G:H]*|H|. The converse family here only shows that a specified divisor-sized subset need not be a subgroup; its witness neither proves nor refutes existence of some subgroup of that order. · mko-lagrange-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

H=\{e,s\}

After

e\in H;\quad ee=e,es=s,se=s,ss=e;\quad e^{-1}=e,s^{-1}=s

Conditions: none

Rule: calc · Uses: G1, S01, S02

Leaf basis: calc

All four products and both inverses stay in H, directly qualifying the subgroup first.



S04 · impliesBefore

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

After

eH=sH=\{e,s\};\quad rH=(rs)H=\{r,rs\};\quad r^2H=(r^2s)H=\{r^2,r^2s\}

Conditions: none

Rule: calc · Uses: G1, S01, S03

Leaf basis: calc

List the left coset for every one of the six g values, then deduplicate to three sets.



S05 · impliesBefore

C_0=\{e,s\},C_1=\{r,rs\},C_2=\{r^2,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: calc · Uses: G1, S02, S03, S04

Leaf basis: calc

The explicit sets are disjoint and cover G; generally, overlap gh1=kh2 gives k^-1*g=h2*h1^-1 in H and hence equality of the cosets.



S06 · impliesBefore

L_g:H\to gH,\quad h\mapsto gh

After

L_g^{-1}(k)=g^{-1}k\Rightarrow |gH|=|H|=2

Conditions: none

Rule: calc · Uses: G1, S02, S03, S04

Leaf basis: calc

The inverse of left translation is explicit, so all three cosets have H's size.



S07 · impliesBefore

G=C_0\sqcup C_1\sqcup C_2,\quad |C_i|=2

After

[G:H]=3,\quad|G|=3\cdot2=6,\quad|H|\mid|G|

Conditions: none

Rule: mko_definition · Uses: G1, S05, S06

Leaf basis: mko_definition

The complete disjoint partition and equal-size certificates yield Lagrange counting.



S08 · impliesBefore

K=\{e,r,s\},\quad|K|=3\mid6

After

rr=r^2\notin K\Rightarrow K\text{ not subgroup};\quad\{e,r,r^2\}\le G

Conditions: none

Rule: calc · Uses: G1, S01, S03, S07

Leaf basis: calc

The specified divisor-sized subset fails closure, while the same group has another order-3 subgroup, preventing a false nonexistence conclusion.







## Substeps and lemmas

No substeps in this guide.





## Reverse audits


- S08 · premise_audit · G1, S01, S03, S07 — Failure refutes K only and not all three-element candidates.
- S07 · premise_audit · G1, S05, S06 — Divisibility comes from actual subgroup cosets, not merely numbers 6 and 2.
- S06 · premise_audit · G1, S02, S03, S04 — g^-1*k lies in H because k lies in gH, and both compositions are identities.
- S05 · premise_audit · G1, S02, S03, S04 — Equal-or-disjoint uses product and inverse closure of H rather than arbitrary grouping.
- S04 · premise_audit · G1, S01, S03 — Listing only eH and rH would miss the two elements in r^2H.
- S03 · premise_audit · G1, S01, S02 — Size 2 dividing 6 is not itself a subgroup certificate; closure is checked here.
- 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.


- K has size 3 dividing 6, so K is a subgroup. → reject — r*r=r^2 is outside K.
- Failure of K proves G has no order-3 subgroup. → reject — The full table qualifies {e,r,r^2} as an order-3 subgroup.
- Two listed cosets suffice for the partition. → reject — r^2 and r^2s would remain uncovered.





## Result and limitations

H={e,s} has three distinct left cosets of size 2, giving 6=[G:H]|H|=3*2.







## Verification and source binding

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


- source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
- adopted source SHA-256: 69a1f66a1aae765143e62e40fb90126333ae697e9503310f59d7ba0e97a5a2b2
- original source SHA-256: 69a1f66a1aae765143e62e40fb90126333ae697e9503310f59d7ba0e97a5a2b2
- 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-lagrange-theorem-finite-groups-01.json)