Library HoTT.Algebra.Groups.QuotientGroup
Require Import Basics Types.
Require Import Truncations.Core.
Require Import Algebra.Congruence.
Require Import Algebra.Groups.Group.
Require Import Algebra.Groups.Subgroup.
Require Export Algebra.Groups.Image.
Require Export Algebra.Groups.Kernel.
Require Export Colimits.Quotient.
Require Import HSet.
Require Import Spaces.Finite.Finite.
Require Import WildCat.
Require Import Modalities.Modality.
Require Import Truncations.Core.
Require Import Algebra.Congruence.
Require Import Algebra.Groups.Group.
Require Import Algebra.Groups.Subgroup.
Require Export Algebra.Groups.Image.
Require Export Algebra.Groups.Kernel.
Require Export Colimits.Quotient.
Require Import HSet.
Require Import Spaces.Finite.Finite.
Require Import WildCat.
Require Import Modalities.Modality.
Local Open Scope mc_scope.
Local Open Scope mc_mult_scope.
Local Open Scope wc_iso_scope.
Section GroupCongruenceQuotient.
A congruence on a group is a relation satisfying R x x' → R y y' → R (x × y) (x' × y'). Because we also require that R is reflexive, we also know that R y y' → R (x × y) (x × y') for any x, and similarly for multiplication on the right by x. We don't need to assume that R is symmetric or transitive.
Definition CongruenceQuotient := G / R.
Global Instance congquot_sgop : SgOp CongruenceQuotient.
Proof.
srapply Quotient_rec2.
- intros x y.
exact (class_of _ (x × y)).
- intros x x' y p. apply qglue. by apply iscong.
- intros x y y' q. apply qglue. by apply iscong.
Defined.
Global Instance congquot_mon_unit : MonUnit CongruenceQuotient.
Proof.
apply class_of, mon_unit.
Defined.
Global Instance congquot_negate : Negate CongruenceQuotient.
Proof.
srapply Quotient_rec.
1: exact (class_of R o negate).
intros x y p; cbn.
symmetry.
rewrite <- (left_identity (-x)).
destruct (left_inverse y).
set (-y × y × -x).
rewrite <- (right_identity (-y)).
destruct (right_inverse x).
unfold g; clear g.
rewrite <- simple_associativity.
apply qglue.
apply iscong; try reflexivity.
apply iscong; try reflexivity.
exact p.
Defined.
Global Instance congquot_sgop_associative : Associative congquot_sgop.
Proof.
srapply Quotient_ind3_hprop; intros x y z.
simpl; by rewrite associativity.
Qed.
Global Instance issemigroup_congquot : IsSemiGroup CongruenceQuotient := {}.
Global Instance congquot_leftidentity
: LeftIdentity congquot_sgop congquot_mon_unit.
Proof.
srapply Quotient_ind_hprop; intro x.
by simpl; rewrite left_identity.
Qed.
Global Instance congquot_rightidentity
: RightIdentity congquot_sgop congquot_mon_unit.
Proof.
srapply Quotient_ind_hprop; intro x.
by simpl; rewrite right_identity.
Qed.
Global Instance ismonoid_quotientgroup : IsMonoid CongruenceQuotient := {}.
Global Instance quotientgroup_leftinverse
: LeftInverse congquot_sgop congquot_negate congquot_mon_unit.
Proof.
srapply Quotient_ind_hprop; intro x.
by simpl; rewrite left_inverse.
Qed.
Global Instance quotientgroup_rightinverse
: RightInverse congquot_sgop congquot_negate congquot_mon_unit.
Proof.
srapply Quotient_ind_hprop; intro x.
by simpl; rewrite right_inverse.
Qed.
Global Instance isgroup_quotientgroup : IsGroup CongruenceQuotient := {}.
End GroupCongruenceQuotient.
Global Instance congquot_sgop : SgOp CongruenceQuotient.
Proof.
srapply Quotient_rec2.
- intros x y.
exact (class_of _ (x × y)).
- intros x x' y p. apply qglue. by apply iscong.
- intros x y y' q. apply qglue. by apply iscong.
Defined.
Global Instance congquot_mon_unit : MonUnit CongruenceQuotient.
Proof.
apply class_of, mon_unit.
Defined.
Global Instance congquot_negate : Negate CongruenceQuotient.
Proof.
srapply Quotient_rec.
1: exact (class_of R o negate).
intros x y p; cbn.
symmetry.
rewrite <- (left_identity (-x)).
destruct (left_inverse y).
set (-y × y × -x).
rewrite <- (right_identity (-y)).
destruct (right_inverse x).
unfold g; clear g.
rewrite <- simple_associativity.
apply qglue.
apply iscong; try reflexivity.
apply iscong; try reflexivity.
exact p.
Defined.
Global Instance congquot_sgop_associative : Associative congquot_sgop.
Proof.
srapply Quotient_ind3_hprop; intros x y z.
simpl; by rewrite associativity.
Qed.
Global Instance issemigroup_congquot : IsSemiGroup CongruenceQuotient := {}.
Global Instance congquot_leftidentity
: LeftIdentity congquot_sgop congquot_mon_unit.
Proof.
srapply Quotient_ind_hprop; intro x.
by simpl; rewrite left_identity.
Qed.
Global Instance congquot_rightidentity
: RightIdentity congquot_sgop congquot_mon_unit.
Proof.
srapply Quotient_ind_hprop; intro x.
by simpl; rewrite right_identity.
Qed.
Global Instance ismonoid_quotientgroup : IsMonoid CongruenceQuotient := {}.
Global Instance quotientgroup_leftinverse
: LeftInverse congquot_sgop congquot_negate congquot_mon_unit.
Proof.
srapply Quotient_ind_hprop; intro x.
by simpl; rewrite left_inverse.
Qed.
Global Instance quotientgroup_rightinverse
: RightInverse congquot_sgop congquot_negate congquot_mon_unit.
Proof.
srapply Quotient_ind_hprop; intro x.
by simpl; rewrite right_inverse.
Qed.
Global Instance isgroup_quotientgroup : IsGroup CongruenceQuotient := {}.
End GroupCongruenceQuotient.
Now we can define the quotient group by a normal subgroup.
Section QuotientGroup.
Context (G : Group) (N : NormalSubgroup G).
Global Instance iscongruence_in_cosetL : IsCongruence (in_cosetL N).
Proof.
srapply Build_IsCongruence.
intros; by apply in_cosetL_cong.
Defined.
Global Instance iscongruence_in_cosetR: IsCongruence (in_cosetR N).
Proof.
srapply Build_IsCongruence.
intros; by apply in_cosetR_cong.
Defined.
Now we have to make a choice whether to pick the left or right cosets. Due to existing convention we shall pick left cosets but we note that we could equally have picked right.
Definition QuotientGroup : Group
:= Build_Group (G / (in_cosetL N)) _ _ _ _.
Definition grp_quotient_map : G $-> QuotientGroup.
Proof.
snrapply Build_GroupHomomorphism.
1: exact (class_of _).
intros ??; reflexivity.
Defined.
Definition grp_quotient_rec {A : Group} (f : G $-> A)
(h : ∀ n : G, N n → f n = mon_unit)
: QuotientGroup $-> A.
Proof.
snrapply Build_GroupHomomorphism.
- srapply Quotient_rec.
+ exact f.
+ cbn; intros x y n.
apply grp_moveR_M1.
rhs_V nrapply (ap (.* f y) (grp_homo_inv _ _)).
rhs_V nrapply grp_homo_op.
symmetry; apply h; assumption.
- intro x.
refine (Quotient_ind_hprop _ _ _).
intro y. revert x.
refine (Quotient_ind_hprop _ _ _).
intro x; simpl.
apply grp_homo_op.
Defined.
Definition grp_quotient_ind_hprop (P : QuotientGroup → Type)
`{∀ x, IsHProp (P x)}
(H1 : ∀ x, P (grp_quotient_map x))
: ∀ x, P x.
Proof.
srapply Quotient_ind_hprop.
exact H1.
Defined.
End QuotientGroup.
Arguments QuotientGroup G N : simpl never.
Arguments grp_quotient_map {_ _}.
Notation "G / N" := (QuotientGroup G N) : group_scope.
:= Build_Group (G / (in_cosetL N)) _ _ _ _.
Definition grp_quotient_map : G $-> QuotientGroup.
Proof.
snrapply Build_GroupHomomorphism.
1: exact (class_of _).
intros ??; reflexivity.
Defined.
Definition grp_quotient_rec {A : Group} (f : G $-> A)
(h : ∀ n : G, N n → f n = mon_unit)
: QuotientGroup $-> A.
Proof.
snrapply Build_GroupHomomorphism.
- srapply Quotient_rec.
+ exact f.
+ cbn; intros x y n.
apply grp_moveR_M1.
rhs_V nrapply (ap (.* f y) (grp_homo_inv _ _)).
rhs_V nrapply grp_homo_op.
symmetry; apply h; assumption.
- intro x.
refine (Quotient_ind_hprop _ _ _).
intro y. revert x.
refine (Quotient_ind_hprop _ _ _).
intro x; simpl.
apply grp_homo_op.
Defined.
Definition grp_quotient_ind_hprop (P : QuotientGroup → Type)
`{∀ x, IsHProp (P x)}
(H1 : ∀ x, P (grp_quotient_map x))
: ∀ x, P x.
Proof.
srapply Quotient_ind_hprop.
exact H1.
Defined.
End QuotientGroup.
Arguments QuotientGroup G N : simpl never.
Arguments grp_quotient_map {_ _}.
Notation "G / N" := (QuotientGroup G N) : group_scope.
Rephrasing that lets you specify the normality proof
Definition QuotientGroup' (G : Group) (N : Subgroup G) (H : IsNormalSubgroup N)
:= QuotientGroup G (Build_NormalSubgroup G N H).
Local Open Scope group_scope.
:= QuotientGroup G (Build_NormalSubgroup G N H).
Local Open Scope group_scope.
Computation rule for grp_quotient_rec.
Corollary grp_quotient_rec_beta `{F : Funext} {G : Group}
(N : NormalSubgroup G) (H : Group)
{A : Group} (f : G $-> A)
(h : ∀ n:G, N n → f n = mon_unit)
: (grp_quotient_rec G N f h) $o grp_quotient_map = f.
Proof.
apply equiv_path_grouphomomorphism; reflexivity.
Defined.
(N : NormalSubgroup G) (H : Group)
{A : Group} (f : G $-> A)
(h : ∀ n:G, N n → f n = mon_unit)
: (grp_quotient_rec G N f h) $o grp_quotient_map = f.
Proof.
apply equiv_path_grouphomomorphism; reflexivity.
Defined.
Computation rule for grp_quotient_rec.
Definition grp_quotient_rec_beta' {G : Group}
(N : NormalSubgroup G) (H : Group)
{A : Group} (f : G $-> A)
(h : ∀ n:G, N n → f n = mon_unit)
: (grp_quotient_rec G N f h) $o grp_quotient_map == f
:= fun _ ⇒ idpath.
(N : NormalSubgroup G) (H : Group)
{A : Group} (f : G $-> A)
(h : ∀ n:G, N n → f n = mon_unit)
: (grp_quotient_rec G N f h) $o grp_quotient_map == f
:= fun _ ⇒ idpath.
The proof of normality is irrelevent up to equivalence. This is unfortunate that it doesn't hold definitionally.
Definition grp_iso_quotient_normal (G : Group) (H : Subgroup G)
{k k' : IsNormalSubgroup H}
: QuotientGroup' G H k ≅ QuotientGroup' G H k'.
Proof.
snrapply Build_GroupIsomorphism'.
1: reflexivity.
intro x.
srapply Quotient_ind_hprop; intro y; revert x.
srapply Quotient_ind_hprop; intro x.
reflexivity.
Defined.
{k k' : IsNormalSubgroup H}
: QuotientGroup' G H k ≅ QuotientGroup' G H k'.
Proof.
snrapply Build_GroupIsomorphism'.
1: reflexivity.
intro x.
srapply Quotient_ind_hprop; intro y; revert x.
srapply Quotient_ind_hprop; intro x.
reflexivity.
Defined.
The universal mapping property for groups
Theorem equiv_grp_quotient_ump {F : Funext} {G : Group} (N : NormalSubgroup G) (H : Group)
: {f : G $-> H & ∀ (n : G), N n → f n = mon_unit}
<~> (G / N $-> H).
Proof.
srapply equiv_adjointify.
- intros [f p].
exact (grp_quotient_rec _ _ f p).
- intro f.
∃ (f $o grp_quotient_map).
intros n h; cbn.
refine (_ @ grp_homo_unit f).
apply ap.
apply qglue; cbn.
rewrite right_identity;
by apply issubgroup_in_inv.
- intros f.
rapply equiv_path_grouphomomorphism.
by srapply Quotient_ind_hprop.
- intros [f p].
srapply path_sigma_hprop; simpl.
exact (grp_quotient_rec_beta N H f p).
Defined.
Section FirstIso.
Context `{Funext} {A B : Group} (phi : A $-> B).
: {f : G $-> H & ∀ (n : G), N n → f n = mon_unit}
<~> (G / N $-> H).
Proof.
srapply equiv_adjointify.
- intros [f p].
exact (grp_quotient_rec _ _ f p).
- intro f.
∃ (f $o grp_quotient_map).
intros n h; cbn.
refine (_ @ grp_homo_unit f).
apply ap.
apply qglue; cbn.
rewrite right_identity;
by apply issubgroup_in_inv.
- intros f.
rapply equiv_path_grouphomomorphism.
by srapply Quotient_ind_hprop.
- intros [f p].
srapply path_sigma_hprop; simpl.
exact (grp_quotient_rec_beta N H f p).
Defined.
Section FirstIso.
Context `{Funext} {A B : Group} (phi : A $-> B).
First we define a map from the quotient by the kernel of phi into the image of phi
Definition grp_image_quotient
: A / grp_kernel phi $-> grp_image phi.
Proof.
srapply grp_quotient_rec.
+ srapply grp_image_in.
+ intros n x.
by apply path_sigma_hprop.
Defined.
: A / grp_kernel phi $-> grp_image phi.
Proof.
srapply grp_quotient_rec.
+ srapply grp_image_in.
+ intros n x.
by apply path_sigma_hprop.
Defined.
The underlying map of this homomorphism is an equivalence
Global Instance isequiv_grp_image_quotient
: IsEquiv grp_image_quotient.
Proof.
snrapply isequiv_surj_emb.
1: srapply cancelR_conn_map.
srapply isembedding_isinj_hset.
refine (Quotient_ind_hprop _ _ _); intro x.
refine (Quotient_ind_hprop _ _ _); intro y.
intros h; simpl in h.
apply qglue; cbn.
apply (equiv_path_sigma_hprop _ _)^-1%equiv in h; cbn in h.
cbn. rewrite grp_homo_op, grp_homo_inv, h.
srapply negate_l.
Defined.
: IsEquiv grp_image_quotient.
Proof.
snrapply isequiv_surj_emb.
1: srapply cancelR_conn_map.
srapply isembedding_isinj_hset.
refine (Quotient_ind_hprop _ _ _); intro x.
refine (Quotient_ind_hprop _ _ _); intro y.
intros h; simpl in h.
apply qglue; cbn.
apply (equiv_path_sigma_hprop _ _)^-1%equiv in h; cbn in h.
cbn. rewrite grp_homo_op, grp_homo_inv, h.
srapply negate_l.
Defined.
First isomorphism theorem for groups
Theorem grp_first_iso : A / grp_kernel phi ≅ grp_image phi.
Proof.
exact (Build_GroupIsomorphism _ _ grp_image_quotient _).
Defined.
End FirstIso.
Proof.
exact (Build_GroupIsomorphism _ _ grp_image_quotient _).
Defined.
End FirstIso.
Quotient groups are finite. Note that we cannot constructively conclude that the normal subgroup H must be finite since G is, therefore we keep it as an assumption.
Global Instance finite_quotientgroup {U : Univalence} (G : Group) (H : NormalSubgroup G)
(fin_G : Finite G) (fin_H : Finite H)
: Finite (QuotientGroup G H).
Proof.
nrapply finite_quotient.
1-5: exact _.
intros x y.
pose (dec_H := detachable_finite_subset H).
apply dec_H.
Defined.
Definition grp_kernel_quotient_iso `{Univalence} {G : Group} (N : NormalSubgroup G)
: GroupIsomorphism N (grp_kernel (@grp_quotient_map G N)).
Proof.
srapply Build_GroupIsomorphism.
- srapply (grp_kernel_corec (subgroup_incl N)).
intro x; cbn.
apply qglue.
apply issubgroup_in_op.
+ exact (issubgroup_in_inv _ x.2).
+ exact issubgroup_in_unit.
- apply isequiv_surj_emb.
2: apply (cancelL_isembedding (g:=pr1)).
intros [g p].
rapply contr_inhabited_hprop.
srefine (tr ((g; _); _)).
+ rewrite <- grp_unit_l, <- negate_mon_unit.
apply (related_quotient_paths (fun x y ⇒ N (-x × y))).
exact p^.
+ srapply path_sigma_hprop.
reflexivity.
Defined.
(fin_G : Finite G) (fin_H : Finite H)
: Finite (QuotientGroup G H).
Proof.
nrapply finite_quotient.
1-5: exact _.
intros x y.
pose (dec_H := detachable_finite_subset H).
apply dec_H.
Defined.
Definition grp_kernel_quotient_iso `{Univalence} {G : Group} (N : NormalSubgroup G)
: GroupIsomorphism N (grp_kernel (@grp_quotient_map G N)).
Proof.
srapply Build_GroupIsomorphism.
- srapply (grp_kernel_corec (subgroup_incl N)).
intro x; cbn.
apply qglue.
apply issubgroup_in_op.
+ exact (issubgroup_in_inv _ x.2).
+ exact issubgroup_in_unit.
- apply isequiv_surj_emb.
2: apply (cancelL_isembedding (g:=pr1)).
intros [g p].
rapply contr_inhabited_hprop.
srefine (tr ((g; _); _)).
+ rewrite <- grp_unit_l, <- negate_mon_unit.
apply (related_quotient_paths (fun x y ⇒ N (-x × y))).
exact p^.
+ srapply path_sigma_hprop.
reflexivity.
Defined.