Lagrange theorem for finite groups

Mathematical statement

The complete Cayley table must certify H as a subgroup. Every left coset gH is equal to or disjoint from every other and all cosets cover G. Left translation h→g*h bijects H with gH, so every coset has size |H|, giving |G|=[G:H]*|H|. The converse family here only shows that a specified divisor-sized subset need not be a subgroup; its witness neither proves nor refutes existence of some subgroup of that order.

Representative formula (exact TeX)

c(G)=i(G,H)*c(H)

Assumptions

• The complete finite Cayley table, identity and candidate subset are explicit.
• Positive families check identity/product/inverse closure and enumerate every coset.

Symbol scope

• c — finite cardinality
• G — finite group
• i — subgroup index
• H — certified subgroup

Intuition, definition, and misconceptions

Left translation is a bijection from H to gH.

Cosets are equal or disjoint and cover G.

The divisibility conclusion supplies no unconditional converse.

A subset size dividing the group order does not certify that subset as a subgroup; a failed candidate is not a certificate that no subgroup of that order exists.

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 is required.
• Natural-number divisibility — Compare group and subgroup orders.
• Cardinality comparison of sets — Count a finite equal-size partition.
• Quotient set — Left cosets form a quotient partition.

Verification boundary

Only complete finite group tables are handled; sampled products do not certify subgroup closure.

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 finite-group Cauchy 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.