Mathlib theorem · Existing formal mathematics
ProofAtlas

Existing in Mathlib · Algebra

Sylow Conjugacy and Counting Theorems

Let p be prime. If a group G has only finitely many Sylow p-subgroups, Mathlib separately proves that any two of them are conjugate and that their number is congruent to 1 modulo p.

Exact theorem

Exact Mathlib statement

theorem card_sylow_modEq_one (p : ℕ) (G : Type*) [Group G] [Fact p.Prime] [Finite (Sylow p G)] : Nat.card (Sylow p G) ≡ 1 [MOD p]

The theorem at a glance

Sylow conjugacy and counting at a glance

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

Mathlib records conjugacy and modulo-p counting in separate declarations: one orbit under G, and a finite Sylow count congruent to one modulo p. Explanatory diagram.
Detailed visual description

The upper theorem spine states the shared prime and finiteness hypotheses. A conjugacy section places several distinct subgroup emblems on one circular G-action orbit, while a separate counting section isolates one fixed subgroup and groups the remaining action orbits into p-divisible packets before the congruence endpoint. A scope footer excludes uniqueness, exact enumeration, and Sylow I.

Statement structure

Statement and scope

Statement map for Sylow Conjugacy and Counting TheoremsThe finite number of Sylow p-subgroups is congruent to 1 modulo p. Claim boundary: This checked artifact is bound only to Mathlib's counting declaration card_sylow_modEq_one: for prime p and a group G with finitely many Sylow p-subgroups, Nat.card (Sylow p G) is congruent to 1 modulo p. The overview mentions the separate upstream instance Sylow.isPretransitive_of_finite only as mathematical context; that conjugacy declaration is not part of this artifact's recorded declaration set. The checked endpoint does not make a Sylow subgroup unique or canonical, give an exact enumeration formula, include the divisibility-by-index clause sometimes grouped into Sylow III, establish existence or containment from Sylow I, or require G itself to be finite. The pinned upstream declaration is card_sylow_modEq_one. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingThe finite number ofSylow p-subgroups iscongruent to 1 modulo p.Claim boundaryThis checked artifact isbound only to Mathlib'scounting declarationcard_sylow_modEq_one:for prime p and a groupG with finitely manySylow p-subgroups,Nat.card (Sylow p G) iscongruent to 1 modulo p.The overview mentionsthe separate upstreaminstanceSylow.isPretransitive_of_finiteonly as mathematicalcontext; that conjugacydeclaration is not partof this artifact'srecorded declarationset. The checkedendpoint does not make aSylow subgroup unique orcanonical, give an exactenumeration formula,include thedivisibility-by-indexclause sometimes groupedinto Sylow III,establish existence orcontainment from SylowI, or require G itselfto be finite.Pinned declarationmathlib ·card_sylow_modEq_oneStatement map for Sylow Conjugacy and Counting TheoremsThe finite number of Sylow p-subgroups is congruent to 1 modulo p. Claim boundary: This checked artifact is bound only to Mathlib's counting declaration card_sylow_modEq_one: for prime p and a group G with finitely many Sylow p-subgroups, Nat.card (Sylow p G) is congruent to 1 modulo p. The overview mentions the separate upstream instance Sylow.isPretransitive_of_finite only as mathematical context; that conjugacy declaration is not part of this artifact's recorded declaration set. The checked endpoint does not make a Sylow subgroup unique or canonical, give an exact enumeration formula, include the divisibility-by-index clause sometimes grouped into Sylow III, establish existence or containment from Sylow I, or require G itself to be finite. The pinned upstream declaration is card_sylow_modEq_one. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingThe finite number ofSylow p-subgroups iscongruent to 1 modulo p.Claim boundaryThis checked artifact isbound only to Mathlib'scounting declarationcard_sylow_modEq_one:for prime p and a groupG with finitely manySylow p-subgroups,Nat.card (Sylow p G) iscongruent to 1 modulo p.The overview mentionsthe separate upstreaminstanceSylow.isPretransitive_of_finiteonly as mathematicalcontext; that conjugacydeclaration is not partof this artifact'srecorded declarationset. The checkedendpoint does not make aSylow subgroup unique orcanonical, give an exactenumeration formula,include thedivisibility-by-indexclause sometimes groupedinto Sylow III,establish existence orcontainment from SylowI, or require G itselfto be finite.Pinned declarationmathlib ·card_sylow_modEq_one

Read the exact Mathlib declaration

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

Conjugation places the Sylow p-subgroups in one equal-status orbit, while a separate fixed-point count leaves one contribution modulo the generic prime p. Explanatory scientific diagram.
Detailed visual description

The left system places six visually identical subgroup tokens directly on one closed directional track, with no center, root, or preferred token. The separate right system uses one token of the same design as a fixed contribution under a self-returning action path. Four nonfixed action rails expose different-sized partial windows of identical tokens and disappear cleanly behind solid portals at both ends, so the pixels do not choose a numerical value of p or display a complete p-element packet. The two systems encode conjugacy and modular counting without asserting a unique Sylow subgroup, an exact enumeration formula, finiteness of G, a Sylow-I construction, or the usual index-divisibility clause.

Why it matters

A mathematical landmark

The second and third Sylow theorems organize maximal p-subgroups both geometrically and arithmetically. These Mathlib generalizations isolate two enduring ideas: conjugation makes the finite Sylow family homogeneous, and a fixed-point count forces its cardinality into the residue class one modulo p.

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 checked artifact is bound only to Mathlib's counting declaration card_sylow_modEq_one: for prime p and a group G with finitely many Sylow p-subgroups, Nat.card (Sylow p G) is congruent to 1 modulo p. The overview mentions the separate upstream instance Sylow.isPretransitive_of_finite only as mathematical context; that conjugacy declaration is not part of this artifact's recorded declaration set. The checked endpoint does not make a Sylow subgroup unique or canonical, give an exact enumeration formula, include the divisibility-by-index clause sometimes grouped into Sylow III, establish existence or containment from Sylow I, or require G itself to be finite.

Source and local evidence

Where the theorem comes from

Existing declaration
card_sylow_modEq_one in mathlib
Relationship
The checked artifact is the upstream declaration itself
Local Lean evidence
artifact.library.mathlib.sylow-second-third-theorems.v001
Source
Open the pinned upstream reference