Finite group

Mathematical statement

Let a finite set G carry a binary operation *. If it is closed on G, associative for every triple, has a two-sided identity and every element has a two-sided inverse, then (G,*) is a finite group. A complete Cayley table allows exhaustive checking of the specified finite model.

Representative formula (exact TeX)

G(S)

Assumptions

• Elements are labeled from 0 to n−1, and the question gives a complete n×n operation table.
• Sampled rows or a few triples do not prove a group. Associativity must be checked over every one of the n³ ordered triples.

Symbol scope

• G — finite-group candidate
• S — finite carrier

Intuition, definition, and misconceptions

Closure means that every table output is still an element of G.

Associativity compares (a*b)*c with a*(b*c) for all triples and is not commutativity.

An identity must work on both sides. An inverse must also multiply to the same identity on both sides.

Do not infer a group from table symmetry or a few examples, or replace two-sided identities and inverses with one-sided conditions.

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

• Set — A group has a fixed carrier set.
• Function mapping — A binary operation must be total and single-valued.
• Binary relations and set projections — A Cayley table assigns an output to each input pair.
• Mathematical proof — Group qualification must close every axiom obligation.

Verification boundary

A complete finite table can exactly audit its specified candidate; no infinite group or unlisted operation is inferred.

not_formalized

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

Next obligation

Add subgroups, homomorphisms, cosets, normal subgroups and quotient 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.