Mathlib theorem · Existing formal mathematics
ProofAtlas

Existing in Mathlib · Algebra

Cauchy's Theorem for Finite Groups

If a prime p divides the number of elements in a finite group G, then G contains at least one element whose order is exactly p.

Exact theorem

Exact Mathlib statement

theorem exists_prime_orderOf_dvd_card {G : Type*} [Group G] [Fintype G] (p : ℕ) [hp : Fact p.Prime] (hdvd : p ∣ Fintype.card G) : ∃ x : G, orderOf x = p

The theorem at a glance

Cauchy's theorem for finite groups at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

Every prime divisor of a finite group's cardinality occurs as the exact order of at least one element. Explanatory diagram.
Detailed visual description

The poster foregrounds the Fintype.card divisibility implication and then follows the checked source route through length-p product-one tuples, cyclic rotation, prime fixed-point counting, a second constant fixed tuple, and extraction of an element of order p. Its footer keeps existence and the selected Fintype endpoint explicit.

Statement structure

From hypotheses to conclusion

Statement map for Cauchy's Theorem for Finite GroupsA prime dividing the cardinality of a finite group is the exact order of some group element. The pinned upstream declaration is exists_prime_orderOf_dvd_card. The exact checked statement is theorem exists_prime_orderOf_dvd_card {G : Type*} [Group G] [Fintype G] (p : ℕ) [hp : Fact p.Prime] (hdvd : p ∣ Fintype.card G) : ∃ x : G, orderOf x = p.Mathematical readingA prime dividing thecardinality of a finitegroup is the exact orderof some group element.Pinned declarationmathlib ·exists_prime_orderOf_dvd_cardExact checked formtheoremexists_prime_orderOf_dvd_card{G : Type*} [Group G][Fintype G] (p : ℕ) [hp: Fact p.Prime] (hdvd :p ∣ Fintype.card G) : ∃x : G, orderOf x = pStatement map for Cauchy's Theorem for Finite GroupsA prime dividing the cardinality of a finite group is the exact order of some group element. The pinned upstream declaration is exists_prime_orderOf_dvd_card. The exact checked statement is theorem exists_prime_orderOf_dvd_card {G : Type*} [Group G] [Fintype G] (p : ℕ) [hp : Fact p.Prime] (hdvd : p ∣ Fintype.card G) : ∃ x : G, orderOf x = p.Mathematical readingA prime dividing thecardinality of a finitegroup is the exact orderof some group element.Pinned declarationmathlib ·exists_prime_orderOf_dvd_cardExact checked formtheoremexists_prime_orderOf_dvd_card{G : Type*} [Group G][Fintype G] (p : ℕ) [hp: Fact p.Prime] (hdvd :p ∣ Fintype.card G) : ∃x : G, orderOf x = p

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

Cyclic rotation of product-one p-tuples and prime fixed-point counting isolate a nonidentity constant tuple whose entry has order p. Explanatory scientific diagram.
Detailed visual description

On a warm technical-paper field, flat rows of circle, diamond, square, star, and triangle tokens represent product-one tuples in S. Cobalt arrows rotate a representative tuple under ρ. A bracket separates representative nonconstant prime cycles from constant fixed rows. At right, one emerald constant row narrows to one witness token and a closed representative orbit, ending at orderOf x = p. Five positions are drawn only as a legible prime example; the indexed theorem is for arbitrary prime p dividing the finite group's cardinality.

Why it matters

A mathematical landmark

Cauchy's theorem reveals how the prime divisors of a finite group's size must appear inside the group's element structure. Mathlib's proof is especially instructive: a cyclic action on product-one tuples converts divisibility into a fixed point and then into an element of exact prime order.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally
Locally reproducedExact upstream declaration replayed
Reviewed pageCurrent public presentation reviewed
Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This page indexes Mathlib's Fintype-cardinality form of Cauchy's theorem for finite groups: for [Group G], [Fintype G], prime p supplied by Fact, and p dividing Fintype.card G, there exists x : G with orderOf x = p. It asserts one existential witness; it does not assert uniqueness, classify the generated subgroup or any other subgroups, or state the nearby apostrophe variant using [Finite G] and Nat.card G.

Source and local evidence

Where the theorem comes from

Existing declaration
exists_prime_orderOf_dvd_card in mathlib
Relationship
The checked artifact is the upstream declaration itself
Local Lean evidence
artifact.library.mathlib.cauchy-theorem-finite-groups.v001
Source
Open the pinned upstream reference