All group elements fixing point zero

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

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)

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

\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}

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

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

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

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

H=\{e,s\}

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

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

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

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

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

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

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

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 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.

• 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

Same-source full JSON (formulas, ASTs, controls and revision log)