# All group elements fixing point zero

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



## Goal and scope

Stab_G(0)={e,s} is a subgroup; Stab_G(3)=G, distinct from orbit and action-kernel constructions.

A point stabilizer in the same complete action, directly checking all products and inverses.





## 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). Points X={0,1,2,3}; (i,j) acts on x<3 by i+(-1)^j*x mod3 and fixes 3.


- G — six-element group carrier
- r,s — labels with r^3=e,s^2=e,sr=r^2s
- H — stabilizer of point zero





## 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 — For the same fully certified action of G on X, Stab_G(x)={g:g*x=x} requires enumeration of every g. Check identity membership, product closure and inverse closure completely in the same group. One fixing element does not give the full stabilizer. A point stabilizer, an orbit and the action kernel are different constructions, even when particular underlying sets happen to agree. · mko-finite-group-stabilizer





## 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

X=\{0,1,2,3\}

After

\begin{array}{c|cccc}\cdot&0&1&2&3\\\hline e&0&1&2&3\\r&1&2&0&3\\r^2&2&0&1&3\\s&0&2&1&3\\rs&1&0&2&3\\r^2s&2&1&0&3\\\end{array}

Conditions: none

Rule: calc · Uses: G1, S02

Leaf basis: calc

Expand all 24 action values. For x<3, g*(h*x)=i+(-1)^j*k+(-1)^(j+l)*x=(gh)*x; both sides fix 3, and e fixes every point.



S04 · impliesBefore

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

After

g\cdot0=0,1,2,0,1,2

Conditions: none

Rule: calc · Uses: G1, S03

Leaf basis: calc

Check the fixed-zero condition for each group element; only e and s give zero.



S05 · impliesBefore

H=\{g:g\cdot0=0\}

After

H=\{e,s\},\quad e\in H

Conditions: none

Rule: mko_definition · Uses: G1, S04

Leaf basis: mko_definition

The complete column gives the whole stabilizer, with explicit identity membership.



S06 · impliesBefore

H=\{e,s\}

After

\begin{array}{c|cc}*&e&s\\\hline e&e&s\\s&s&e\end{array}

Conditions: none

Rule: calc · Uses: G1, S01, S05

Leaf basis: calc

All four products in H stay in H, directly checking closure rather than sampling only s^2.



S07 · impliesBefore

e^{-1}=e,\quad s^{-1}=s

After

H^{-1}=H\Rightarrow H\le G

Conditions: none

Rule: calc · Uses: G1, S02, S05, S06

Leaf basis: calc

Both inverses lie in H; with identity and product closure this qualifies H as a subgroup.



S08 · impliesBefore

g\cdot3=3\quad\forall g\in G

After

\operatorname{Stab}_G(3)=G\ne H

Conditions: none

Rule: calc · Uses: G1, S03, S05

Leaf basis: calc

The complete 3-column gives a different point stabilizer, showing point dependence.



S09 · impliesBefore

s\cdot1=2,\quad rs\cdot0=1,\quad r^2s\cdot0=2,\quad r\cdot0=1,\quad r^2\cdot0=2

After

\{g:g\cdot x=x\ \forall x\in X\}=\{e\}\ne\operatorname{Stab}_G(0)

Conditions: none

Rule: calc · Uses: G1, S03, S05, S08

Leaf basis: calc

Explicitly exclude all five other elements from fixing every point; the action kernel is {e}, whereas the point stabilizer contains s.







## Substeps and lemmas

No substeps in this guide.





## Reverse audits


- S09 · premise_audit · G1, S03, S05, S08 — The action kernel fixes every point, while the stabilizer fixes the specified 0; the quantifiers differ.
- S08 · premise_audit · G1, S03, S05 — The stabilizer size at 3 cannot be substituted for the one at 0.
- S07 · premise_audit · G1, S02, S05, S06 — The subgroup conclusion is directly checked, not inferred from size two.
- S06 · premise_audit · G1, S01, S05 — Use the same parent multiplication, not a new operation on H.
- S05 · premise_audit · G1, S04 — H is a subset of G, while orbit 0 is a subset of point carrier X.
- S04 · premise_audit · G1, S03 — Finding e is not enough; nonidentity s fixes zero too.
- S03 · premise_audit · G1, S02 — Every group-point pair has one value; the formula covers all 144 triples, not merely permutation rows.
- 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.


- Stab(0)={e} after checking only identity. → reject — s*0=0 also holds.
- Stab(0)=orbit 0. → reject — The former is group elements {e,s}; the latter points {0,1,2}.
- A point stabilizer is the action kernel. → reject — s fixes 0 but sends 1 to 2 and is not in the action kernel.





## Result and limitations

Stab_G(0)={e,s} is a subgroup; Stab_G(3)=G, distinct from orbit and action-kernel constructions.







## Verification and source binding

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


- source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
- adopted source SHA-256: 7a379848dfe73153d408591aeb7867aa3fea39f96d117af769783f52c3e4b936
- original source SHA-256: 7a379848dfe73153d408591aeb7867aa3fea39f96d117af769783f52c3e4b936
- Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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