Cauchy theorem for finite groups

Mathematical statement

A complete Cayley table must certify G as a finite group, and p must be prime and divide |G|. There exists g with g^p=e while g^k≠e for every1≤k<p, hence ord(g)=p. Complete finite enumeration here supplies witnesses for a specified model, not a replacement for the general theorem proof. Do not extend the prime-divisor conclusion to every composite divisor d of |G|.

Representative formula (exact TeX)

o(g)=p

Assumptions

• Positive families use complete closed finite Cayley tables passing associativity, identity and inverse checks.
• Element order is the least positive identity-return exponent, and every smaller positive power is checked.

Symbol scope

• o — element order
• g — finite-group element
• p — prime divisor of the group order

Intuition, definition, and misconceptions

For every prime divisor there is at least one exact-order-p witness.

An order-p certificate checks the p-th power and every smaller positive power.

Exhausting a finite table certifies only that group; the general Cauchy theorem remains a theorem-level obligation.

g^p=e alone does not establish exact order p; a prime-divisor conclusion cannot be copied to arbitrary composite divisors.

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

• Lagrange theorem for finite groups — The generated cyclic-subgroup order divides the group order.
• Prime natural number — p must qualify as prime.
• Finite group — The group operation is complete.
• Finite group action — General proofs may use actions and orbit counting.

Verification boundary

Finite models do not replace the general proof, and not every composite divisor need be an element order.

not_formalized

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

Next obligation

Add Sylow theorems and finite-group structure 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.