Kernel of a finite group homomorphism

Mathematical statement

Both groups require complete Cayley-table qualification. f is a total mapping and must satisfy f(x*y)=f(x)*f(y) for every pair. Its kernel is the entire target-identity preimage, with subgroup and all-g conjugation closure checked directly. f(a)=f(b) exactly when a^(-1)*b belongs to the kernel, so each nonempty fiber is a kernel coset. The kernel is the source-identity singleton exactly when f is injective. Nonhomomorphisms cannot invoke these theorems; quotient/image isomorphism is not claimed here.

Representative formula (exact TeX)

K(f)=H

Assumptions

• Families declare two complete finite group tables and a total mapping array.
• Kernel, fibers, normality and injectivity are completely enumerated in the finite model.

Symbol scope

• K — kernel operator
• f — finite group homomorphism
• H — normal kernel subgroup

Intuition, definition, and misconceptions

Homomorphism qualification precedes kernel conclusions.

Audit kernel normality with complete conjugation.

Fiber/coset equivalence and trivial-kernel injectivity are separate checks.

The identity preimage of an arbitrary total map is not automatically a normal kernel, and G/kernel-image isomorphism is not established merely by naming the kernel.

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

• Subgroups and group homomorphisms — Use the homomorphism law and kernel background.
• Finite normal subgroup — Kernel normality has its own obligation.
• Function mapping — The map is complete and total.

Verification boundary

This slice focuses on normality, fibers and injectivity rather than repeating generic kernel/image exercises; the first isomorphism theorem remains later work.

not_formalized

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

Next obligation

Add the first isomorphism theorem for finite groups.

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.