Finite normal subgroup

Mathematical statement

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.

Representative formula (exact TeX)

N(H,G)=1

Assumptions

• Families use complete finite Cayley tables, an identity and an explicit subset, not sampled multiplication.
• Conjugate sets, left/right cosets and inverses are completely enumerated from the same table.

Symbol scope

• N — normality predicate
• H — candidate subgroup
• G — finite group

Intuition, definition, and misconceptions

Subgroup and normal-subgroup qualification are different obligations.

Full conjugate-set equality and all-g left/right coset equality are equivalent checks.

One escaping conjugate element refutes normality.

Neither finiteness nor a subset size dividing |G| makes it normal; a few sampled g values are not enough.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Finite group — Use the complete Cayley table.
• Subgroups and group homomorphisms — Subgroup closure comes first.
• Quantifiers and explicit domains — Normality has an all-g obligation.

Verification boundary

Only complete finite group models are handled; quotient operations and the first isomorphism theorem are not materialized here.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add finite quotient groups, kernels and the first isomorphism theorem.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.