Normality of rotations from every conjugate

Goal and scope

N={e,r,r2} is normal; H={e,s} is a subgroup but is not normal.

Complete subgroup, all-g conjugation and left/right coset checks; quotient multiplication is not constructed here.

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
• N,H — normal and nonnormal subgroups

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 G, and H must first pass identity, product and inverse closure. Normality then requires gHg^(-1)=H for every g in G, equivalently gH=Hg for every g. One explicit g,h with h in H but g*h*g^(-1) outside H refutes normality. Trivial and whole subgroups are normal. Finiteness, sampled conjugates or generator checks without a proved generation obligation are insufficient. Well-defined quotient multiplication is not yet claimed here. · mko-finite-normal-subgroup

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)

N=\{e,r,r^2\}

\begin{array}{c|ccc}*&e&r&r^2\\\hline e&e&r&r^2\\r&r&r^2&e\\r^2&r^2&e&r\\\end{array}

e^{-1}=e,\quad r^{-1}=r^2,\quad(r^2)^{-1}=r

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

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

\begin{array}{c|ccc}gng^{-1}&e&r&r^2\\\hline e&e&r&r^2\\r&e&r&r^2\\r^2&e&r&r^2\\s&e&r^2&r\\rs&e&r^2&r\\r^2s&e&r^2&r\\\end{array}

N=\{e,r,r^2\},\quad B=\{s,rs,r^2s\}

gN=Ng=N\ (g=e,r,r^2);\qquad gN=Ng=B\ (g=s,rs,r^2s)

N\le G,\quad\forall g:\ gNg^{-1}=N

N\trianglelefteq G

H=\{e,s\}

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

g=r,\quad h=s\in H

r s r^{-1}=r^2s\notin H\Rightarrow H\not\trianglelefteq G

g\in G

g\{e\}g^{-1}=\{e\},\quad gGg^{-1}=G

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S10 · premise_audit · G1, S02, S07, S09 — These boundaries do not make every other subgroup normal.
• S09 · premise_audit · G1, S01, S02, S08 — r inverse is r^2 from the same table; product order cannot be reversed.
• S08 · premise_audit · G1, S01, S02 — The upcoming witness refutes normality, not H's subgroup status.
• S07 · premise_audit · G1, S04, S05, S06 — No quotient table or first-isomorphism theorem is claimed here.
• S06 · premise_audit · G1, S01, S04, S05 — Each coset uses all three parent-table products, not a single representative image.
• S05 · premise_audit · G1, S01, S02, S04 — All six g values are checked rather than testing r,s without a generation proof.
• S04 · premise_audit · G1, S02, S03 — Normality has not yet been proved; finiteness or an index number alone is not used to upgrade it.
• S03 · premise_audit · G1, S01 — Subgroup closure must be checked first; size 3 alone is insufficient.
• 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.

• H has size 2 dividing 6, hence normal. → reject — r*s*r inverse=r^2s lies outside H.
• Product closure of N alone proves normality. → reject — Closure addresses subgroup status; all conjugations remain necessary, as subgroup H is not normal.
• Testing conjugation by identity suffices. → reject — Identity conjugation fixes every subset and cannot detect H's r witness.

Result and limitations

N={e,r,r2} is normal; H={e,s} is a subgroup but is not normal.

Verification and source binding

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

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: afda0b4ee1c3617c0aaa2344f6db7818aaff3418cd954b96fa712b07765a328e
• original source SHA-256: afda0b4ee1c3617c0aaa2344f6db7818aaff3418cd954b96fa712b07765a328e
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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