First isomorphism theorem for finite groups

Mathematical statement

Qualify both complete Cayley tables and the complete total homomorphism f first. Construct the full kernel, image and kernel-coset quotient. The induced map bar_f(g*kernel)=f(g) must agree on every representative of every coset, preserve every product and be injective and surjective onto the image. Equal cardinalities alone do not prove an isomorphism. If f is not surjective, do not replace the image by the entire codomain. Finite enumeration certifies the specified model, not the general abstract theorem; a nonhomomorphism cannot invoke the theorem.

Representative formula (exact TeX)

I(f)=h

Assumptions

• Families use two complete finite group tables and an aligned total mapping array.
• The quotient and image carriers are constructed by complete finite enumeration.

Symbol scope

• I — first-isomorphism construction
• f — finite group homomorphism
• h — induced group isomorphism

Intuition, definition, and misconceptions

Qualify the homomorphism, kernel, image and quotient first.

Representative independence precedes the induced-map conclusion.

Check the full homomorphism and bijection, retaining the image/codomain boundary.

Equal cardinalities alone do not certify isomorphism, and a nonsurjective f does not identify G/kernel with the entire target group.

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

• Kernel of a finite group homomorphism — Use kernel and fiber cosets.
• Finite quotient group — Require the full quotient carrier and well-defined multiplication.
• Subgroups and group homomorphisms — The image forms a subgroup of the target.
• Bijectivity and unique preimages — Check injectivity and surjectivity of the induced map.

Verification boundary

These complete finite-enumeration certificates do not replace the general abstract proof of first isomorphism.

not_formalized

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

Next obligation

Add group-center and class-equation structural consequences.

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.