module Algebra.Group.Subgroup where
private variable ℓ ℓ' : Level G : Group ℓ private module _ {ℓ} where open Cat.Displayed.Instances.Subobjects (Groups ℓ) public
Subgroups🔗
A subgroup of a group is a monomorphism that is, an object of the poset of subobjects Since group homomorphisms are injective exactly when their underlying function is an embedding, we can alternatively describe this as a condition on a predicate
Subgroup : Group ℓ → Type (lsuc ℓ) Subgroup {ℓ = ℓ} G = Subobject G
A proposition
of a group
represents a subgroup if it contains the group unit
, is closed under
multiplication, and is closed under inverses.
record represents-subgroup (G : Group ℓ) (H : ℙ ⌞ G ⌟) : Type ℓ where open Group-on (G .snd) field has-unit : unit ∈ H has-⋆ : ∀ {x y} → x ∈ H → y ∈ H → (x ⋆ y) ∈ H has-inv : ∀ {x} → x ∈ H → x ⁻¹ ∈ H
If represents a subgroup, then its total space inherits a group structure from and the first projection is a group homomorphism.
rep-subgroup→group-on : (H : ℙ ⌞ G ⌟) → represents-subgroup G H → Group-on (Σ[ x ∈ G ] x ∈ H) rep-subgroup→group-on {G = G} H sg = to-group-on sg' where open Group-on (G .snd) open represents-subgroup sg sg' : make-group (Σ[ x ∈ G ] x ∈ H) sg' .make-group.group-is-set = hlevel 2 sg' .make-group.unit = unit , has-unit sg' .make-group.mul (x , x∈) (y , y∈) = x ⋆ y , has-⋆ x∈ y∈ sg' .make-group.inv (x , x∈) = x ⁻¹ , has-inv x∈ sg' .make-group.assoc x y z = Σ-prop-path! associative sg' .make-group.invl x = Σ-prop-path! inversel sg' .make-group.idl x = Σ-prop-path! idl predicate→subgroup : (H : ℙ ⌞ G ⌟) → represents-subgroup G H → Subgroup G predicate→subgroup {G = G} H p = record { map = it ; monic = ism } where it : Groups.Hom (el! (Σ _ (∣_∣ ⊙ H)) , rep-subgroup→group-on H p) G it .hom = fst it .preserves .is-group-hom.pres-⋆ x y = refl ism : Groups.is-monic it ism = Homomorphism-monic it λ p → Σ-prop-path! p
Kernels and images🔗
To a group homomorphism we can associate two canonical subgroups, one of and one of image factorisation, written is the subgroup of “reachable by mapping through ”, and kernel, written is the subgroup of which sends to the unit.
The kernel can be cheapily described as a limit: It is the equaliser of and the zero morphism — which, recall, is the unique map which breaks down as
module _ {ℓ} where open Canonical-kernels (Groups ℓ) ∅ᴳ Groups-equalisers public Ker-subgroup : ∀ {A B : Group ℓ} → Groups.Hom A B → Subgroup A Ker-subgroup f = record { map = kernel ; monic = is-equaliser→is-monic _ has-is-kernel } where open Kernel (Ker f)
Every group homomorphism
has an image factorisation
defined by equipping its set-theoretic image
with a group structure
inherited from
More concretely, we can describe the elements of
as the “mere fibres” of
They consist of a point
together with (the truncation of) a fibre of
over
We multiply
(in the fibre over
with
(in the fibre over
giving the element
in the fibre over
module _ {ℓ} {A B : Group ℓ} (f : Groups.Hom A B) where private module A = Group-on (A .snd) module B = Group-on (B .snd) module f = is-group-hom (f .preserves) Tpath : {x y : image (apply f)} → x .fst ≡ y .fst → x ≡ y Tpath {x} {y} p = Σ-prop-path! p abstract Tset : is-set (image (apply f)) Tset = hlevel 2 module Kerf = Kernel (Ker f)
For reasons that will become clear later, we denote the image of when regarded as its own group, by and reserve the notation for that group regarded as a subgroup of .
The construction of a group structure on
is unsurprising, so we leave it in this <details>
tag
for the curious reader.
T : Type ℓ T = image (apply f) A/ker[_] : Group ℓ A/ker[_] = to-group grp where unit : T unit = B.unit , inc (A.unit , f.pres-id) inv : T → T inv (x , p) = x B.⁻¹ , ∥-∥-map (λ { (y , p) → y A.⁻¹ , f.pres-inv ∙ ap B._⁻¹ p }) p mul : T → T → T mul (x , xp) (y , yp) = x B.⋆ y , ∥-∥-elim₂ (λ _ _ → squash) (λ { (x* , xp) (y* , yp) → inc (x* A.⋆ y* , f.pres-⋆ _ _ ∙ ap₂ B._⋆_ xp yp) }) xp yp grp : make-group T grp .make-group.group-is-set = Tset grp .make-group.unit = unit grp .make-group.mul = mul grp .make-group.inv = inv grp .make-group.assoc = λ x y z → Tpath B.associative grp .make-group.invl = λ x → Tpath B.inversel grp .make-group.idl = λ x → Tpath B.idl
That the canonical inclusion map
deserves the name “image” comes from
breaking down as a (regular) epimorphism into
(written A→im
), followed by that
map:
A→im : Groups.Hom A A/ker[_] A→im .hom x = f # x , inc (x , refl) A→im .preserves .is-group-hom.pres-⋆ x y = Tpath (f.pres-⋆ _ _) im→B : Groups.Hom A/ker[_] B im→B .hom (b , _) = b im→B .preserves .is-group-hom.pres-⋆ x y = refl
When this monomorphism is taken as primary, we refer to as
Im[_] : Subgroup B Im[_] = record { map = im→B ; monic = im↪B } where im↪B : Groups.is-monic im→B im↪B = Homomorphism-monic im→B Tpath
The first isomorphism theorem🔗
The reason for denoting the set-theoretic image of (which is a subobject of , equipped with group operation) by is the first isomorphism theorem (though we phrase it more categorically): The image of serves as a quotient for (the congruence generated by)
In more classical texts, the first isomorphism theorem is phrased in terms of two pre-existing objects (defined as the set of cosets of regarded as a subgroup) and (defined as above). Here we have opted for a more categorical phrasing of that theorem: We know what the universal property of is — namely that it is a specific colimit — so the specific construction used to implement it does not matter.
1st-iso-theorem : is-coequaliser (Groups ℓ) (Zero.zero→ ∅ᴳ) Kerf.kernel A→im 1st-iso-theorem = coeq where open Groups open is-coequaliser module Ak = Group-on (A/ker[_] .snd)
More specifically, in a diagram like the one below, the indicated dotted arrow always exists and is unique, witnessing that the map is a coequaliser (hence that it is a regular epi, as we mentioned above).
The condition placed on is that This means that it, like sends everything in to zero (this is the defining property of Note that in the code below we do not elide the zero composite
elim : ∀ {F} {e' : Groups.Hom A F} (p : e' Groups.∘ Zero.zero→ ∅ᴳ ≡ e' Groups.∘ Kerf.kernel) → ∀ {x : ⌞ B ⌟} → ∥ fibre (apply f) x ∥ → _ elim {F = F} {e' = e'} p {x} = ∥-∥-rec-set (F .snd .Group-on.has-is-set) ((e' #_) ⊙ fst) const where abstract module e' = is-group-hom (e' .preserves) module F = Group-on (F .snd)
To eliminate from under a propositional truncation, we must prove that the map is constant when thought of as a map In other words, it means that is “independent of the choice of representative”. This follows from algebraic manipulation of group homomorphisms + the assumed identity
const' : ∀ (x y : fibre (apply f) x) → e' # (x .fst) F.— e' # (y .fst) ≡ F.unit const' (y , q) (z , r) = (e' # y) F.— (e' # z) ≡˘⟨ e'.pres-diff ⟩≡˘ e' # (y A.— z) ≡⟨ happly (sym (ap hom p)) (y A.— z , aux) ⟩≡ e' # A.unit ≡⟨ e'.pres-id ⟩≡ F.unit ∎ where
This assumption allows us to reduce “show that is constant on a specific subset” to “show that when ”; But that’s just algebra, hence uninteresting:
aux : f # (y A.— z) ≡ B.unit aux = f # (y A.— z) ≡⟨ f.pres-diff ⟩≡ f # y B.— f # z ≡⟨ ap₂ B._—_ q r ⟩≡ x B.— x ≡⟨ B.inverser ⟩≡ B.unit ∎ const : ∀ (x y : fibre (apply f) x) → e' # (x .fst) ≡ e' # (y .fst) const a b = F.zero-diff (const' a b)
The rest of the construction is almost tautological: By
definition, if
then
so the quotient map
does indeed coequalise
and
As a final word on the rest of the construction, most of it is applying
induction (∥-∥-elim
and friends) so
that our colimiting map elim
will compute.
coeq : is-coequaliser (Groups ℓ) (Zero.zero→ ∅ᴳ) Kerf.kernel A→im coeq .coequal = ext λ x p → f.pres-id ∙ sym p coeq .universal {F = F} {e' = e'} p = gh where module F = Group-on (F .snd) module e' = is-group-hom (e' .preserves) gh : Groups.Hom _ _ gh .hom (x , t) = elim {e' = e'} p t gh .preserves .is-group-hom.pres-⋆ (x , q) (y , r) = ∥-∥-elim₂ {P = λ q r → elim p (((x , q) Ak.⋆ (y , r)) .snd) ≡ elim p q F.⋆ elim p r} (λ _ _ → F.has-is-set _ _) (λ x y → e'.pres-⋆ _ _) q r coeq .factors = Grp↪Sets-is-faithful refl coeq .unique {F} {p = p} {colim = colim} prf = ext λ x y p → ap# colim (Σ-prop-path! (sym p)) ∙ happly (ap hom prf) y
Representing kernels🔗
If an evil wizard kidnaps your significant others and demands that you find out whether a predicate is a kernel, how would you go about doing it? Well, I should point out that no matter how evil the wizard is, they are still human: The predicate definitely represents a subgroup, in the sense introduced above — so there’s definitely a group homomorphism All we need to figure out is whether there exists a group and a map such that as subgroups of
We begin by assuming that we have a kernel and investigating some properties that the fibres of its inclusion have. Of course, the fibre over is inhabited, and they are closed under multiplication and inverses, though we shall not make note of that here.
module _ {ℓ} {A B : Group ℓ} (f : Groups.Hom A B) where private module Ker[f] = Kernel (Ker f) module f = is-group-hom (f .preserves) module A = Group-on (A .snd) module B = Group-on (B .snd) kerf : ⌞ Ker[f].ker ⌟ → ⌞ A ⌟ kerf = Ker[f].kernel .hom has-zero : fibre kerf A.unit has-zero = (A.unit , f.pres-id) , refl has-⋆ : ∀ {x y} → fibre kerf x → fibre kerf y → fibre kerf (x A.⋆ y) has-⋆ ((a , p) , q) ((b , r) , s) = (a A.⋆ b , f.pres-⋆ _ _ ·· ap₂ B._⋆_ p r ·· B.idl) , ap₂ A._⋆_ q s
It turns out that is also closed under conjugation by elements of the enveloping group, in that if (quickly switching to “multiplicative” notation for the unit), then must be as well: for we have
has-conjugate : ∀ {x y} → fibre kerf x → fibre kerf (y A.⋆ x A.⋆ y A.⁻¹) has-conjugate {x} {y} ((a , p) , q) = (_ , path) , refl where path = f # (y A.⋆ (x A.— y)) ≡⟨ ap (f #_) A.associative ⟩≡ f # ((y A.⋆ x) A.— y) ≡⟨ f.pres-diff ⟩≡ ⌜ f # (y A.⋆ x) ⌝ B.— f # y ≡⟨ ap₂ B._—_ (f.pres-⋆ y x) refl ⟩≡ ⌜ f # y B.⋆ f # x ⌝ B.— f # y ≡⟨ ap₂ B._—_ (ap (_ B.⋆_) (ap (f #_) (sym q) ∙ p) ∙ B.idr) refl ⟩≡ f # y B.— f # y ≡˘⟨ f.pres-diff ⟩≡˘ f # (y A.— y) ≡⟨ ap (f #_) A.inverser ∙ f.pres-id ⟩≡ B.unit ∎
It turns out that this last property is enough to pick out exactly the kernels amongst the representations of subgroups: If is closed under conjugation, then generates an equivalence relation on the set underlying (namely, and equip the quotient of this equivalence relation with a group structure. The kernel of the quotient map is then We call a predicate representing a kernel a normal subgroup, and we denote this in shorthand by
record normal-subgroup (G : Group ℓ) (H : ℙ ⌞ G ⌟) : Type ℓ where open Group-on (G .snd) field has-rep : represents-subgroup G H has-conjugate : ∀ {x y} → x ∈ H → (y ⋆ x ⋆ y ⁻¹) ∈ H has-conjugatel : ∀ {x y} → y ∈ H → ((x ⋆ y) ⋆ x ⁻¹) ∈ H has-conjugatel yin = subst (_∈ H) associative (has-conjugate yin) has-comm : ∀ {x y} → (x ⋆ y) ∈ H → (y ⋆ x) ∈ H has-comm {x = x} {y} mem = subst (_∈ H) p (has-conjugate mem) where p = x ⁻¹ ⋆ ⌜ (x ⋆ y) ⋆ x ⁻¹ ⁻¹ ⌝ ≡˘⟨ ap¡ associative ⟩≡˘ x ⁻¹ ⋆ x ⋆ y ⋆ ⌜ x ⁻¹ ⁻¹ ⌝ ≡⟨ ap! inv-inv ⟩≡ x ⁻¹ ⋆ x ⋆ y ⋆ x ≡⟨ associative ⟩≡ (x ⁻¹ ⋆ x) ⋆ y ⋆ x ≡⟨ ap₂ _⋆_ inversel refl ∙ idl ⟩≡ y ⋆ x ∎ open represents-subgroup has-rep public
So, suppose we have a normal subgroup We define the underlying type of the quotient to be the quotient of the relation It can be equipped with a group operation inherited from but this is incredibly tedious to do.
module _ (Grp : Group ℓ) {H : ℙ ⌞ Grp ⌟} (N : normal-subgroup Grp H) where open normal-subgroup N open Group-on (Grp .snd) renaming (inverse to inv) private G0 = ⌞ Grp ⌟ rel : G0 → G0 → Type _ rel x y = (x ⋆ y ⁻¹) ∈ H rel-refl : ∀ x → rel x x rel-refl _ = subst (_∈ H) (sym inverser) has-unit
G/H : Type _ G/H = G0 / rel op : G/H → G/H → G/H op = Quot-op₂ rel-refl rel-refl _⋆_ (λ w x y z a b → rem₃ y z w x b a) where
To prove that the group operation _⋆_
descends to the
quotient, we prove that it takes related inputs to related outputs — a
characterisation of binary operations on quotients we can invoke since
the relation we’re quotienting by is reflexive. It suffices to show that
whenever
and
are both in
which is a tedious but straightforward calculation:
module _ (w x y z : G0) (w-x∈ : (w ⋆ inv x) ∈ H) (y-z∈ : (y ⋆ inv z) ∈ H) where abstract rem₁ : ((w — x) ⋆ (inv z ⋆ y)) ∈ H rem₁ = has-⋆ w-x∈ (has-comm y-z∈) rem₂ : ((w ⋆ (inv x — z)) ⋆ y) ∈ H rem₂ = subst (_∈ H) (associative ∙ ap (_⋆ y) (sym associative)) rem₁ rem₃ : ((y ⋆ w) — (z ⋆ x)) ∈ H rem₃ = subst (_∈ H) (associative ∙ ap₂ _⋆_ refl (sym inv-comm)) (has-comm rem₂)
To define inverses on the quotient, it suffices to show that whenever we also have
inverse : G/H → G/H inverse = Coeq-rec (λ x → inc (inv x)) λ { (x , y , r) → quot (p x y r) } where abstract p : ∀ x y → (x — y) ∈ H → (inv x — inv y) ∈ H p x y r = has-comm (subst (_∈ H) inv-comm (has-inv r))
Even after this tedious algebra, it still remains to show that the operation is associative and has inverses. Fortunately, since equality in a group is a proposition, these follow from the group axioms on rather directly:
Group-on-G/H : make-group G/H Group-on-G/H .make-group.group-is-set = squash Group-on-G/H .make-group.unit = inc unit Group-on-G/H .make-group.mul = op Group-on-G/H .make-group.inv = inverse Group-on-G/H .make-group.assoc = elim! λ x y z → ap Coeq.inc associative Group-on-G/H .make-group.invl = elim! λ x → ap Coeq.inc inversel Group-on-G/H .make-group.idl = elim! λ x → ap Coeq.inc idl infix 25 _/ᴳ_ _/ᴳ_ : Group _ _/ᴳ_ = to-group Group-on-G/H incl : Groups.Hom Grp _/ᴳ_ incl .hom = inc incl .preserves .is-group-hom.pres-⋆ x y = refl
Before we show that the kernel of the quotient map is isomorphic to the subgroup we started with (and indeed, that this isomorphism commutes with the respective, so that they determine the same subobject of we must show that the relation is an equivalence relation; We can then appeal to effectivity of quotients to conclude that, if then
private rel-sym : ∀ {x y} → rel x y → rel y x rel-sym h = subst (_∈ H) (inv-comm ∙ ap (_⋆ _) inv-inv) (has-inv h) rel-trans : ∀ {x y z} → rel x y → rel y z → rel x z rel-trans {x} {y} {z} h g = subst (_∈ H) p (has-⋆ h g) where p = (x — y) ⋆ (y — z) ≡˘⟨ associative ⟩≡˘ x ⋆ ⌜ y ⁻¹ ⋆ (y — z) ⌝ ≡⟨ ap! associative ⟩≡ x ⋆ ⌜ (y ⁻¹ ⋆ y) — z ⌝ ≡⟨ ap! (ap (_⋆ _) inversel ∙ idl) ⟩≡ x — z ∎ open Congruence normal-subgroup→congruence : Congruence _ _ normal-subgroup→congruence ._∼_ = rel normal-subgroup→congruence .has-is-prop x y = hlevel 1 normal-subgroup→congruence .reflᶜ = rel-refl _ normal-subgroup→congruence ._∙ᶜ_ = rel-trans normal-subgroup→congruence .symᶜ = rel-sym /ᴳ-effective : ∀ {x y} → Path G/H (inc x) (inc y) → rel x y /ᴳ-effective = effective normal-subgroup→congruence
private module Ker[incl] = Kernel (Ker incl) Ker-sg = Ker-subgroup incl H-sg = predicate→subgroup H has-rep H-g = H-sg .domain
The two halves of the isomorphism are now very straightforward to define: If we have then by effectivity, and by the group laws. Conversely, if then thus they are identified in the quotient. Thus, the predicate recovers the subgroup And (the total space of) that predicate is exactly the kernel of
Ker[incl]≅H-group : Ker[incl].ker Groups.≅ H-g Ker[incl]≅H-group = Groups.make-iso to from il ir where to : Groups.Hom _ _ to .hom (x , p) = x , subst (_∈ H) (ap (_ ⋆_) inv-unit ∙ idr) x-0∈H where x-0∈H = /ᴳ-effective p to .preserves .is-group-hom.pres-⋆ _ _ = Σ-prop-path! refl from : Groups.Hom _ _ from .hom (x , p) = x , quot (subst (_∈ H) (sym idr ∙ ap (_ ⋆_) (sym inv-unit)) p) from .preserves .is-group-hom.pres-⋆ _ _ = Σ-prop-path! refl il = ext λ x x∈H → Σ-prop-path! refl ir = ext λ x x∈H → Σ-prop-path! refl
To show that these are equal as subgroups of we must show that the isomorphism above commutes with the inclusions; But this is immediate by computation, so we can conclude: Every normal subgroup is a kernel.
Ker[incl]≡H↪G : Ker-sg ≡ H-sg Ker[incl]≡H↪G = done where open Precategory (Sub Grp) open Groups._≅_ Ker[incl]≅H-group ker≤H : Ker-sg ≤ₘ H-sg ker≤H .map = to ker≤H .sq = Grp↪Sets-is-faithful refl H≤ker : H-sg ≤ₘ Ker-sg H≤ker .map = from H≤ker .sq = Grp↪Sets-is-faithful refl done = Sub-is-category Groups-is-category .to-path (Sub-antisym ker≤H H≤ker)