Mathlib theorem · Existing formal mathematics
ProofAtlas

Existing in Mathlib · Algebra

First Sylow Theorem

Let p be prime. Whenever pⁿ divides the number of elements in a finite group G, there is at least one subgroup K with exactly pⁿ elements.

Exact theorem

Exact Mathlib statement

theorem Sylow.exists_subgroup_card_pow_prime {G : Type u} [Group G] [Finite G] (p : ℕ) {n : ℕ} [Fact p.Prime] (hdvd : p ^ n ∣ Nat.card G) : ∃ K : Subgroup G, Nat.card K = p ^ n

The theorem at a glance

First Sylow Theorem at a glance

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

Every prime power dividing a finite group order occurs as the exact order of a subgroup. Explanatory diagram.
Detailed visual description

The poster centers the exact prime-power implication, then follows fixed cosets through a normalizer quotient to an order-p direction that lifts and enlarges a subgroup one rung at a time. Its footer keeps the existential boundary explicit.

Statement structure

Statement and scope

Statement map for First Sylow TheoremIf a prime power divides the order of a finite group, the group has a subgroup of exactly that prime-power order. Claim boundary: This target indexes Mathlib's subgroup-existence theorem for every prime-power divisor of a finite group's cardinality. It asserts one existential witness; it does not assert uniqueness, normality, conjugacy, a subgroup count, or the second and third Sylow theorems. The pinned upstream declaration is Sylow.exists_subgroup_card_pow_prime. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingIf a prime power dividesthe order of a finitegroup, the group has asubgroup of exactly thatprime-power order.Claim boundaryThis target indexesMathlib'ssubgroup-existencetheorem for everyprime-power divisor of afinite group'scardinality. It assertsone existential witness;it does not assertuniqueness, normality,conjugacy, a subgroupcount, or the second andthird Sylow theorems.Pinned declarationmathlib ·Sylow.exists_subgroup_card_pow_primeStatement map for First Sylow TheoremIf a prime power divides the order of a finite group, the group has a subgroup of exactly that prime-power order. Claim boundary: This target indexes Mathlib's subgroup-existence theorem for every prime-power divisor of a finite group's cardinality. It asserts one existential witness; it does not assert uniqueness, normality, conjugacy, a subgroup count, or the second and third Sylow theorems. The pinned upstream declaration is Sylow.exists_subgroup_card_pow_prime. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingIf a prime power dividesthe order of a finitegroup, the group has asubgroup of exactly thatprime-power order.Claim boundaryThis target indexesMathlib'ssubgroup-existencetheorem for everyprime-power divisor of afinite group'scardinality. It assertsone existential witness;it does not assertuniqueness, normality,conjugacy, a subgroupcount, or the second andthird Sylow theorems.Pinned declarationmathlib ·Sylow.exists_subgroup_card_pow_prime

Read the exact Mathlib declaration

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

A prime-power divisor of a finite group order is realized by the cardinality of a subgroup. Explanatory diagram.
Detailed visual description

A broad finite group is rendered as a field of discrete elements. One noncanonical witness route grows through nested subgroup boundaries by one factor of p at a time, ending at a highlighted subgroup of cardinality pⁿ. Several faint alternative subgroup contours prevent the witness from appearing unique.

Why it matters

A mathematical landmark

Sylow's theorems organize the prime-power structure hidden inside finite groups. Mathlib's selected declaration gives the strong existence form for every prime-power divisor, not only the largest power of p dividing the group 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 target indexes Mathlib's subgroup-existence theorem for every prime-power divisor of a finite group's cardinality. It asserts one existential witness; it does not assert uniqueness, normality, conjugacy, a subgroup count, or the second and third Sylow theorems.

Source and local evidence

Where the theorem comes from

Existing declaration
Sylow.exists_subgroup_card_pow_prime in mathlib
Relationship
The checked artifact is the upstream declaration itself
Local Lean evidence
artifact.library.mathlib.first-sylow-theorem.v001
Source
Open the pinned upstream reference