# Defining a group given by generators and relations #

Given a subset rels of relations of the free group on a type α, this file constructs the group given by generators x : α and relations r ∈ rels.

## Main definitions #

• PresentedGroup rels: the quotient group of the free group on a type α by a subset rels of relations of the free group on α.
• of: The canonical map from α to a presented group with generators α.
• toGroup f: the canonical group homomorphism PresentedGroup rels → G, given a function f : α → G from a type α to a group G which satisfies the relations rels.

## Tags #

generators, relations, group presentations

def PresentedGroup {α : Type u_1} (rels : Set (FreeGroup α)) :
Type u_1

Given a set of relations, rels, over a type α, PresentedGroup constructs the group with generators x : α and relations rels as a quotient of FreeGroup α.

Equations
Instances For
instance PresentedGroup.instGroup {α : Type u_1} (rels : Set (FreeGroup α)) :
Equations
def PresentedGroup.of {α : Type u_1} {rels : Set (FreeGroup α)} (x : α) :

of is the canonical map from α to a presented group with generators x : α. The term x is mapped to the equivalence class of the image of x in FreeGroup α.

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Instances For
@[simp]
theorem PresentedGroup.closure_range_of {α : Type u_1} (rels : Set (FreeGroup α)) :
Subgroup.closure (Set.range PresentedGroup.of) =

The generators of a presented group generate the presented group. That is, the subgroup closure of the set of generators equals ⊤.

theorem PresentedGroup.closure_rels_subset_ker {α : Type u_1} {G : Type u_2} [] {f : αG} {rels : Set (FreeGroup α)} (h : rrels, (FreeGroup.lift f) r = 1) :
(FreeGroup.lift f).ker
theorem PresentedGroup.to_group_eq_one_of_mem_closure {α : Type u_1} {G : Type u_2} [] {f : αG} {rels : Set (FreeGroup α)} (h : rrels, (FreeGroup.lift f) r = 1) (x : ) :
x (FreeGroup.lift f) x = 1
def PresentedGroup.toGroup {α : Type u_1} {G : Type u_2} [] {f : αG} {rels : Set (FreeGroup α)} (h : rrels, (FreeGroup.lift f) r = 1) :

The extension of a map f : α → G that satisfies the given relations to a group homomorphism from PresentedGroup rels → G.

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Instances For
@[simp]
theorem PresentedGroup.toGroup.of {α : Type u_1} {G : Type u_2} [] {f : αG} {rels : Set (FreeGroup α)} (h : rrels, (FreeGroup.lift f) r = 1) {x : α} :
= f x
theorem PresentedGroup.toGroup.unique {α : Type u_1} {G : Type u_2} [] {f : αG} {rels : Set (FreeGroup α)} (h : rrels, (FreeGroup.lift f) r = 1) (g : PresentedGroup rels →* G) (hg : ∀ (x : α), g = f x) {x : PresentedGroup rels} :
g x =
theorem PresentedGroup.ext_iff {α : Type u_1} {G : Type u_2} [] {rels : Set (FreeGroup α)} {φ : PresentedGroup rels →* G} {ψ : PresentedGroup rels →* G} :
φ = ψ ∀ (x : α), φ = ψ
theorem PresentedGroup.ext {α : Type u_1} {G : Type u_2} [] {rels : Set (FreeGroup α)} {φ : PresentedGroup rels →* G} {ψ : PresentedGroup rels →* G} (hx : ∀ (x : α), φ = ψ ) :
φ = ψ
def PresentedGroup.equivPresentedGroup {α : Type u_1} {β : Type u_3} (rels : Set (FreeGroup α)) (e : α β) :

Presented groups of isomorphic types are isomorphic.

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Instances For
theorem PresentedGroup.equivPresentedGroup_apply_of {α : Type u_1} {β : Type u_3} (x : α) (rels : Set (FreeGroup α)) (e : α β) :
theorem PresentedGroup.equivPresentedGroup_symm_apply_of {α : Type u_1} {β : Type u_3} (x : β) (rels : Set (FreeGroup α)) (e : α β) :
.symm = PresentedGroup.of (e.symm x)
instance PresentedGroup.instInhabited {α : Type u_1} (rels : Set (FreeGroup α)) :
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• = { default := 1 }