Quotients of groups by normal subgroups

This files develops the basic theory of quotients of groups by normal subgroups. In particular it proves Noether's first and second isomorphism theorems.

Main definitions

• mk': the canonical group homomorphism G →* G/N given a normal subgroup N of G.
• lift φ: the group homomorphism G/N →* H given a group homomorphism φ : G →* H such that N ⊆ ker φ.
• map f: the group homomorphism G/N →* H/M given a group homomorphism f : G →* H such that N ⊆ f⁻¹(M).

Main statements

• QuotientGroup.quotientKerEquivRange: Noether's first isomorphism theorem, an explicit isomorphism G/ker φ → range φ for every group homomorphism φ : G →* H.
• QuotientGroup.quotientInfEquivProdNormalQuotient: Noether's second isomorphism theorem, an explicit isomorphism between H/(H ∩ N) and (HN)/N given a subgroup H and a normal subgroup N of a group G.
• QuotientGroup.quotientQuotientEquivQuotient: Noether's third isomorphism theorem, the canonical isomorphism between (G / N) / (M / N) and G / M, where N ≤ M.

Tags

isomorphism theorems, quotient groups

def QuotientAddGroup.con {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] : AddCon G

The additive congruence relation generated by a normal additive subgroup.

Equations

• QuotientAddGroup.con N = { toSetoid := QuotientAddGroup.leftRel N, add' := ⋯ }

theorem QuotientAddGroup.con.proof_1 {G : Type u_1} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] (a : G) (b : G) (c : G) (d : G) (hab : Setoid.r a b) (hcd : Setoid.r c d) : Setoid.r (a + c) (b + d)

def QuotientGroup.con {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] : Con G

The congruence relation generated by a normal subgroup.

Equations

• QuotientGroup.con N = { toSetoid := QuotientGroup.leftRel N, mul' := ⋯ }

instance QuotientAddGroup.Quotient.addGroup {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] : AddGroup (G ⧸ N)

Equations

• QuotientAddGroup.Quotient.addGroup N = AddCon.addGroup (QuotientAddGroup.con N)

instance QuotientGroup.Quotient.group {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] : Group (G ⧸ N)

Equations

• QuotientGroup.Quotient.group N = Con.group (QuotientGroup.con N)

theorem QuotientAddGroup.mk'.proof_1 {G : Type u_1} [AddGroup G] (N : AddSubgroup G) : ∀ (x x_1 : G), ↑(x + x_1) = ↑(x + x_1)

def QuotientAddGroup.mk' {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] : G →+ G ⧸ N

The additive group homomorphism from G to G/N.

Equations

• QuotientAddGroup.mk' N = AddMonoidHom.mk' QuotientAddGroup.mk ⋯

def QuotientGroup.mk' {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] : G →* G ⧸ N

The group homomorphism from G to G/N.

Equations

• QuotientGroup.mk' N = MonoidHom.mk' QuotientGroup.mk ⋯

@[simp]

theorem QuotientAddGroup.coe_mk' {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] : ⇑(QuotientAddGroup.mk' N) = QuotientAddGroup.mk

@[simp]

theorem QuotientGroup.coe_mk' {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] : ⇑(QuotientGroup.mk' N) = QuotientGroup.mk

@[simp]

theorem QuotientAddGroup.mk'_apply {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] (x : G) : (QuotientAddGroup.mk' N) x = ↑x

@[simp]

theorem QuotientGroup.mk'_apply {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] (x : G) : (QuotientGroup.mk' N) x = ↑x

theorem QuotientAddGroup.mk'_surjective {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] : Function.Surjective ⇑(QuotientAddGroup.mk' N)

theorem QuotientGroup.mk'_surjective {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] : Function.Surjective ⇑(QuotientGroup.mk' N)

theorem QuotientAddGroup.mk'_eq_mk' {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] {x : G} {y : G} : (QuotientAddGroup.mk' N) x = (QuotientAddGroup.mk' N) y ↔ ∃ z ∈ N, x + z = y

theorem QuotientGroup.mk'_eq_mk' {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] {x : G} {y : G} : (QuotientGroup.mk' N) x = (QuotientGroup.mk' N) y ↔ ∃ z ∈ N, x * z = y

theorem QuotientAddGroup.sound {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] (U : Set (G ⧸ N)) (g : ↥(AddSubgroup.op N)) : g +ᵥ ⇑(QuotientAddGroup.mk' N) ⁻¹' U = ⇑(QuotientAddGroup.mk' N) ⁻¹' U

theorem QuotientGroup.sound {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] (U : Set (G ⧸ N)) (g : ↥(Subgroup.op N)) : g • ⇑(QuotientGroup.mk' N) ⁻¹' U = ⇑(QuotientGroup.mk' N) ⁻¹' U

theorem QuotientAddGroup.addMonoidHom_ext {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] {M : Type x} [AddMonoid M] ⦃f : G ⧸ N →+ M⦄ ⦃g : G ⧸ N →+ M⦄ (h : AddMonoidHom.comp f (QuotientAddGroup.mk' N) = AddMonoidHom.comp g (QuotientAddGroup.mk' N)) : f = g

Two AddMonoidHoms from an additive quotient group are equal if their compositions with AddQuotientGroup.mk' are equal.

See note [partially-applied ext lemmas].

theorem QuotientGroup.monoidHom_ext {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] {M : Type x} [Monoid M] ⦃f : G ⧸ N →* M⦄ ⦃g : G ⧸ N →* M⦄ (h : MonoidHom.comp f (QuotientGroup.mk' N) = MonoidHom.comp g (QuotientGroup.mk' N)) : f = g

Two MonoidHoms from a quotient group are equal if their compositions with QuotientGroup.mk' are equal.

See note [partially-applied ext lemmas].

@[simp]

theorem QuotientAddGroup.eq_zero_iff {G : Type u} [AddGroup G] {N : AddSubgroup G} [nN : AddSubgroup.Normal N] (x : G) : ↑x = 0 ↔ x ∈ N

@[simp]

theorem QuotientGroup.eq_one_iff {G : Type u} [Group G] {N : Subgroup G} [nN : Subgroup.Normal N] (x : G) : ↑x = 1 ↔ x ∈ N

abbrev QuotientAddGroup.ker_le_range_iff.match_1 {H : Type u_1} [AddGroup H] {I : Type u_2} [AddGroup I] (g : H →+ I) (motive : ↥(AddMonoidHom.ker g) → Prop) : ∀ (x : ↥(AddMonoidHom.ker g)), (∀ (val : H) (hx : val ∈ AddMonoidHom.ker g), motive { val := val, property := hx }) → motive x

Equations

• ⋯ = ⋯

theorem QuotientAddGroup.ker_le_range_iff {G : Type u} [AddGroup G] {H : Type v} [AddGroup H] {I : Type w} [AddGroup I] (f : G →+ H) [AddSubgroup.Normal (AddMonoidHom.range f)] (g : H →+ I) : AddMonoidHom.ker g ≤ AddMonoidHom.range f ↔ AddMonoidHom.comp (QuotientAddGroup.mk' (AddMonoidHom.range f)) (AddSubgroup.subtype (AddMonoidHom.ker g)) = 0

theorem QuotientGroup.ker_le_range_iff {G : Type u} [Group G] {H : Type v} [Group H] {I : Type w} [Group I] (f : G →* H) [Subgroup.Normal (MonoidHom.range f)] (g : H →* I) : MonoidHom.ker g ≤ MonoidHom.range f ↔ MonoidHom.comp (QuotientGroup.mk' (MonoidHom.range f)) (Subgroup.subtype (MonoidHom.ker g)) = 1

@[simp]

theorem QuotientAddGroup.ker_mk' {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] : AddMonoidHom.ker (QuotientAddGroup.mk' N) = N

@[simp]

theorem QuotientGroup.ker_mk' {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] : MonoidHom.ker (QuotientGroup.mk' N) = N

theorem QuotientAddGroup.eq_iff_sub_mem {G : Type u} [AddGroup G] {N : AddSubgroup G} [nN : AddSubgroup.Normal N] {x : G} {y : G} : ↑x = ↑y ↔ x - y ∈ N

theorem QuotientGroup.eq_iff_div_mem {G : Type u} [Group G] {N : Subgroup G} [nN : Subgroup.Normal N] {x : G} {y : G} : ↑x = ↑y ↔ x / y ∈ N

instance QuotientAddGroup.Quotient.addCommGroup {G : Type u_1} [AddCommGroup G] (N : AddSubgroup G) : AddCommGroup (G ⧸ N)

Equations

• QuotientAddGroup.Quotient.addCommGroup N = AddCommGroup.mk ⋯

theorem QuotientAddGroup.Quotient.addCommGroup.proof_1 {G : Type u_1} [AddCommGroup G] (N : AddSubgroup G) (a : G ⧸ N) (b : G ⧸ N) : a + b = b + a

instance QuotientGroup.Quotient.commGroup {G : Type u_1} [CommGroup G] (N : Subgroup G) : CommGroup (G ⧸ N)

Equations

• QuotientGroup.Quotient.commGroup N = CommGroup.mk ⋯

@[simp]

theorem QuotientAddGroup.mk_zero {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] : ↑0 = 0

@[simp]

theorem QuotientGroup.mk_one {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] : ↑1 = 1

@[simp]

theorem QuotientAddGroup.mk_add {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] (a : G) (b : G) : ↑(a + b) = ↑a + ↑b

@[simp]

theorem QuotientGroup.mk_mul {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] (a : G) (b : G) : ↑(a * b) = ↑a * ↑b

@[simp]

theorem QuotientAddGroup.mk_neg {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] (a : G) : ↑(-a) = -↑a

@[simp]

theorem QuotientGroup.mk_inv {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] (a : G) : ↑a⁻¹ = (↑a)⁻¹

@[simp]

theorem QuotientAddGroup.mk_sub {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] (a : G) (b : G) : ↑(a - b) = ↑a - ↑b

@[simp]

theorem QuotientGroup.mk_div {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] (a : G) (b : G) : ↑(a / b) = ↑a / ↑b

@[simp]

theorem QuotientAddGroup.mk_nsmul {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] (a : G) (n : ℕ) : ↑(n • a) = n • ↑a

@[simp]

theorem QuotientGroup.mk_pow {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] (a : G) (n : ℕ) : ↑(a ^ n) = ↑a ^ n

@[simp]

theorem QuotientAddGroup.mk_zsmul {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] (a : G) (n : ℤ) : ↑(n • a) = n • ↑a

@[simp]

theorem QuotientGroup.mk_zpow {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] (a : G) (n : ℤ) : ↑(a ^ n) = ↑a ^ n

@[simp]

theorem QuotientAddGroup.mk_sum {G : Type u_1} {ι : Type u_2} [AddCommGroup G] (N : AddSubgroup G) (s : Finset ι) {f : ι → G} : ↑(Finset.sum s f) = Finset.sum s fun (i : ι) => ↑(f i)

@[simp]

theorem QuotientGroup.mk_prod {G : Type u_1} {ι : Type u_2} [CommGroup G] (N : Subgroup G) (s : Finset ι) {f : ι → G} : ↑(Finset.prod s f) = Finset.prod s fun (i : ι) => ↑(f i)

@[simp]

theorem QuotientAddGroup.map_mk'_self {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] : AddSubgroup.map (QuotientAddGroup.mk' N) N = ⊥

@[simp]

theorem QuotientGroup.map_mk'_self {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] : Subgroup.map (QuotientGroup.mk' N) N = ⊥

theorem QuotientAddGroup.lift.proof_1 {G : Type u_1} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] {M : Type u_2} [AddMonoid M] (φ : G →+ M) (HN : N ≤ AddMonoidHom.ker φ) (x : G) (y : G) (h : (QuotientAddGroup.con N) x y) : (AddCon.ker φ) x y

def QuotientAddGroup.lift {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] {M : Type x} [AddMonoid M] (φ : G →+ M) (HN : N ≤ AddMonoidHom.ker φ) : G ⧸ N →+ M

An AddGroup homomorphism φ : G →+ M with N ⊆ ker(φ) descends (i.e. lifts) to a group homomorphism G/N →* M.

Equations

• QuotientAddGroup.lift N φ HN = AddCon.lift (QuotientAddGroup.con N) φ ⋯

def QuotientGroup.lift {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] {M : Type x} [Monoid M] (φ : G →* M) (HN : N ≤ MonoidHom.ker φ) : G ⧸ N →* M

A group homomorphism φ : G →* M with N ⊆ ker(φ) descends (i.e. lifts) to a group homomorphism G/N →* M.

Equations

• QuotientGroup.lift N φ HN = Con.lift (QuotientGroup.con N) φ ⋯

@[simp]

theorem QuotientAddGroup.lift_mk {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] {M : Type x} [AddMonoid M] {φ : G →+ M} (HN : N ≤ AddMonoidHom.ker φ) (g : G) : (QuotientAddGroup.lift N φ HN) ↑g = φ g

@[simp]

theorem QuotientGroup.lift_mk {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] {M : Type x} [Monoid M] {φ : G →* M} (HN : N ≤ MonoidHom.ker φ) (g : G) : (QuotientGroup.lift N φ HN) ↑g = φ g

@[simp]

theorem QuotientAddGroup.lift_mk' {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] {M : Type x} [AddMonoid M] {φ : G →+ M} (HN : N ≤ AddMonoidHom.ker φ) (g : G) : (QuotientAddGroup.lift N φ HN) ↑g = φ g

@[simp]

theorem QuotientGroup.lift_mk' {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] {M : Type x} [Monoid M] {φ : G →* M} (HN : N ≤ MonoidHom.ker φ) (g : G) : (QuotientGroup.lift N φ HN) ↑g = φ g

@[simp]

theorem QuotientAddGroup.lift_quot_mk {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] {M : Type x} [AddMonoid M] {φ : G →+ M} (HN : N ≤ AddMonoidHom.ker φ) (g : G) : (QuotientAddGroup.lift N φ HN) (Quot.mk Setoid.r g) = φ g

@[simp]

theorem QuotientGroup.lift_quot_mk {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] {M : Type x} [Monoid M] {φ : G →* M} (HN : N ≤ MonoidHom.ker φ) (g : G) : (QuotientGroup.lift N φ HN) (Quot.mk Setoid.r g) = φ g

def QuotientAddGroup.map {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] {H : Type v} [AddGroup H] (M : AddSubgroup H) [AddSubgroup.Normal M] (f : G →+ H) (h : N ≤ AddSubgroup.comap f M) : G ⧸ N →+ H ⧸ M

An AddGroup homomorphism f : G →+ H induces a map G/N →+ H/M if N ⊆ f⁻¹(M).

Equations

• QuotientAddGroup.map N M f h = QuotientAddGroup.lift N (AddMonoidHom.comp (QuotientAddGroup.mk' M) f) ⋯

theorem QuotientAddGroup.map.proof_1 {G : Type u_1} [AddGroup G] (N : AddSubgroup G) {H : Type u_2} [AddGroup H] (M : AddSubgroup H) (f : G →+ H) (h : N ≤ AddSubgroup.comap f M) ⦃x : G⦄ (hx : x ∈ N) : ↑(f x) = ↑0

def QuotientGroup.map {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] {H : Type v} [Group H] (M : Subgroup H) [Subgroup.Normal M] (f : G →* H) (h : N ≤ Subgroup.comap f M) : G ⧸ N →* H ⧸ M

A group homomorphism f : G →* H induces a map G/N →* H/M if N ⊆ f⁻¹(M).

Equations

• QuotientGroup.map N M f h = QuotientGroup.lift N (MonoidHom.comp (QuotientGroup.mk' M) f) ⋯

@[simp]

theorem QuotientAddGroup.map_mk {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] {H : Type v} [AddGroup H] (M : AddSubgroup H) [AddSubgroup.Normal M] (f : G →+ H) (h : N ≤ AddSubgroup.comap f M) (x : G) : (QuotientAddGroup.map N M f h) ↑x = ↑(f x)

@[simp]

theorem QuotientGroup.map_mk {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] {H : Type v} [Group H] (M : Subgroup H) [Subgroup.Normal M] (f : G →* H) (h : N ≤ Subgroup.comap f M) (x : G) : (QuotientGroup.map N M f h) ↑x = ↑(f x)

theorem QuotientAddGroup.map_mk' {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] {H : Type v} [AddGroup H] (M : AddSubgroup H) [AddSubgroup.Normal M] (f : G →+ H) (h : N ≤ AddSubgroup.comap f M) (x : G) : (QuotientAddGroup.map N M f h) ((QuotientAddGroup.mk' N) x) = ↑(f x)

theorem QuotientGroup.map_mk' {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] {H : Type v} [Group H] (M : Subgroup H) [Subgroup.Normal M] (f : G →* H) (h : N ≤ Subgroup.comap f M) (x : G) : (QuotientGroup.map N M f h) ((QuotientGroup.mk' N) x) = ↑(f x)

theorem QuotientAddGroup.map_id_apply {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] (h : optParam (N ≤ AddSubgroup.comap (AddMonoidHom.id G) N) ⋯) (x : G ⧸ N) : (QuotientAddGroup.map N N (AddMonoidHom.id G) h) x = x

theorem QuotientGroup.map_id_apply {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] (h : optParam (N ≤ Subgroup.comap (MonoidHom.id G) N) ⋯) (x : G ⧸ N) : (QuotientGroup.map N N (MonoidHom.id G) h) x = x

@[simp]

theorem QuotientAddGroup.map_id {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] (h : optParam (N ≤ AddSubgroup.comap (AddMonoidHom.id G) N) ⋯) : QuotientAddGroup.map N N (AddMonoidHom.id G) h = AddMonoidHom.id (G ⧸ N)

@[simp]

theorem QuotientGroup.map_id {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] (h : optParam (N ≤ Subgroup.comap (MonoidHom.id G) N) ⋯) : QuotientGroup.map N N (MonoidHom.id G) h = MonoidHom.id (G ⧸ N)

@[simp]

theorem QuotientAddGroup.map_map {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] {H : Type v} [AddGroup H] {I : Type u_1} [AddGroup I] (M : AddSubgroup H) (O : AddSubgroup I) [AddSubgroup.Normal M] [AddSubgroup.Normal O] (f : G →+ H) (g : H →+ I) (hf : N ≤ AddSubgroup.comap f M) (hg : M ≤ AddSubgroup.comap g O) (hgf : optParam (N ≤ AddSubgroup.comap (AddMonoidHom.comp g f) O) ⋯) (x : G ⧸ N) : (QuotientAddGroup.map M O g hg) ((QuotientAddGroup.map N M f hf) x) = (QuotientAddGroup.map N O (AddMonoidHom.comp g f) hgf) x

@[simp]

theorem QuotientGroup.map_map {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] {H : Type v} [Group H] {I : Type u_1} [Group I] (M : Subgroup H) (O : Subgroup I) [Subgroup.Normal M] [Subgroup.Normal O] (f : G →* H) (g : H →* I) (hf : N ≤ Subgroup.comap f M) (hg : M ≤ Subgroup.comap g O) (hgf : optParam (N ≤ Subgroup.comap (MonoidHom.comp g f) O) ⋯) (x : G ⧸ N) : (QuotientGroup.map M O g hg) ((QuotientGroup.map N M f hf) x) = (QuotientGroup.map N O (MonoidHom.comp g f) hgf) x

@[simp]

theorem QuotientAddGroup.map_comp_map {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] {H : Type v} [AddGroup H] {I : Type u_1} [AddGroup I] (M : AddSubgroup H) (O : AddSubgroup I) [AddSubgroup.Normal M] [AddSubgroup.Normal O] (f : G →+ H) (g : H →+ I) (hf : N ≤ AddSubgroup.comap f M) (hg : M ≤ AddSubgroup.comap g O) (hgf : optParam (N ≤ AddSubgroup.comap (AddMonoidHom.comp g f) O) ⋯) : AddMonoidHom.comp (QuotientAddGroup.map M O g hg) (QuotientAddGroup.map N M f hf) = QuotientAddGroup.map N O (AddMonoidHom.comp g f) hgf

@[simp]

theorem QuotientGroup.map_comp_map {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] {H : Type v} [Group H] {I : Type u_1} [Group I] (M : Subgroup H) (O : Subgroup I) [Subgroup.Normal M] [Subgroup.Normal O] (f : G →* H) (g : H →* I) (hf : N ≤ Subgroup.comap f M) (hg : M ≤ Subgroup.comap g O) (hgf : optParam (N ≤ Subgroup.comap (MonoidHom.comp g f) O) ⋯) : MonoidHom.comp (QuotientGroup.map M O g hg) (QuotientGroup.map N M f hf) = QuotientGroup.map N O (MonoidHom.comp g f) hgf

@[simp]

theorem QuotientAddGroup.image_coe {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] : QuotientAddGroup.mk '' ↑N = 0

@[simp]

theorem QuotientGroup.image_coe {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] : QuotientGroup.mk '' ↑N = 1

theorem QuotientAddGroup.preimage_image_coe {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] (s : Set G) : QuotientAddGroup.mk ⁻¹' (QuotientAddGroup.mk '' s) = ↑N + s

theorem QuotientGroup.preimage_image_coe {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] (s : Set G) : QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' s) = ↑N * s

theorem QuotientAddGroup.image_coe_inj {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] {s : Set G} {t : Set G} : QuotientAddGroup.mk '' s = QuotientAddGroup.mk '' t ↔ ↑N + s = ↑N + t

theorem QuotientGroup.image_coe_inj {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] {s : Set G} {t : Set G} : QuotientGroup.mk '' s = QuotientGroup.mk '' t ↔ ↑N * s = ↑N * t

theorem QuotientAddGroup.congr.proof_3 {G : Type u_1} [AddGroup G] {H : Type u_2} [AddGroup H] (G' : AddSubgroup G) (H' : AddSubgroup H) (e : G ≃+ H) (he : AddSubgroup.map (↑e) G' = H') : G' ≤ AddSubgroup.comap (↑e) H'

def QuotientAddGroup.congr {G : Type u} [AddGroup G] {H : Type v} [AddGroup H] (G' : AddSubgroup G) (H' : AddSubgroup H) [AddSubgroup.Normal G'] [AddSubgroup.Normal H'] (e : G ≃+ H) (he : AddSubgroup.map (↑e) G' = H') : G ⧸ G' ≃+ H ⧸ H'

QuotientAddGroup.congr lifts the isomorphism e : G ≃ H to G ⧸ G' ≃ H ⧸ H', given that e maps G to H.

Equations

• One or more equations did not get rendered due to their size.

theorem QuotientAddGroup.congr.proof_7 {G : Type u_2} [AddGroup G] {H : Type u_1} [AddGroup H] (G' : AddSubgroup G) (H' : AddSubgroup H) [AddSubgroup.Normal G'] [AddSubgroup.Normal H'] (e : G ≃+ H) (he : AddSubgroup.map (↑e) G' = H') (x : H ⧸ H') : (QuotientAddGroup.map G' H' ↑e ⋯) ((QuotientAddGroup.map H' G' ↑(AddEquiv.symm e) ⋯) x) = x

theorem QuotientAddGroup.congr.proof_5 {G : Type u_2} [AddGroup G] {H : Type u_1} [AddGroup H] (G' : AddSubgroup G) (H' : AddSubgroup H) (e : G ≃+ H) (he : AddSubgroup.map (↑e) G' = H') : H' ≤ AddSubgroup.comap (↑(AddEquiv.symm e)) G'

theorem QuotientAddGroup.congr.proof_1 {G : Type u_2} [AddGroup G] {H : Type u_1} [AddGroup H] : AddMonoidHomClass (G ≃+ H) G H

theorem QuotientAddGroup.congr.proof_8 {G : Type u_1} [AddGroup G] {H : Type u_2} [AddGroup H] (G' : AddSubgroup G) (H' : AddSubgroup H) [AddSubgroup.Normal G'] [AddSubgroup.Normal H'] (e : G ≃+ H) (he : AddSubgroup.map (↑e) G' = H') (x : G ⧸ G') (y : G ⧸ G') : (QuotientAddGroup.map G' H' ↑e ⋯).toFun (x + y) = (QuotientAddGroup.map G' H' ↑e ⋯).toFun x + (QuotientAddGroup.map G' H' ↑e ⋯).toFun y

theorem QuotientAddGroup.congr.proof_6 {G : Type u_1} [AddGroup G] {H : Type u_2} [AddGroup H] (G' : AddSubgroup G) (H' : AddSubgroup H) [AddSubgroup.Normal G'] [AddSubgroup.Normal H'] (e : G ≃+ H) (he : AddSubgroup.map (↑e) G' = H') (x : G ⧸ G') : (QuotientAddGroup.map H' G' ↑(AddEquiv.symm e) ⋯) ((QuotientAddGroup.map G' H' ↑e ⋯) x) = x

theorem QuotientAddGroup.congr.proof_2 {G : Type u_1} [AddGroup G] {H : Type u_2} [AddGroup H] (G' : AddSubgroup G) (H' : AddSubgroup H) (e : G ≃+ H) (he : AddSubgroup.map (↑e) G' = H') : G' ≤ AddSubgroup.comap (↑e) H'

theorem QuotientAddGroup.congr.proof_4 {G : Type u_1} [AddGroup G] {H : Type u_2} [AddGroup H] : AddMonoidHomClass (H ≃+ G) H G

def QuotientGroup.congr {G : Type u} [Group G] {H : Type v} [Group H] (G' : Subgroup G) (H' : Subgroup H) [Subgroup.Normal G'] [Subgroup.Normal H'] (e : G ≃* H) (he : Subgroup.map (↑e) G' = H') : G ⧸ G' ≃* H ⧸ H'

QuotientGroup.congr lifts the isomorphism e : G ≃ H to G ⧸ G' ≃ H ⧸ H', given that e maps G to H.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem QuotientGroup.congr_mk {G : Type u} [Group G] {H : Type v} [Group H] (G' : Subgroup G) (H' : Subgroup H) [Subgroup.Normal G'] [Subgroup.Normal H'] (e : G ≃* H) (he : Subgroup.map (↑e) G' = H') (x : G) : (QuotientGroup.congr G' H' e he) ↑x = ↑(e x)

theorem QuotientGroup.congr_mk' {G : Type u} [Group G] {H : Type v} [Group H] (G' : Subgroup G) (H' : Subgroup H) [Subgroup.Normal G'] [Subgroup.Normal H'] (e : G ≃* H) (he : Subgroup.map (↑e) G' = H') (x : G) : (QuotientGroup.congr G' H' e he) ((QuotientGroup.mk' G') x) = (QuotientGroup.mk' H') (e x)

@[simp]

theorem QuotientGroup.congr_apply {G : Type u} [Group G] {H : Type v} [Group H] (G' : Subgroup G) (H' : Subgroup H) [Subgroup.Normal G'] [Subgroup.Normal H'] (e : G ≃* H) (he : Subgroup.map (↑e) G' = H') (x : G) : (QuotientGroup.congr G' H' e he) ↑x = (QuotientGroup.mk' H') (e x)

@[simp]

theorem QuotientGroup.congr_refl {G : Type u} [Group G] (G' : Subgroup G) [Subgroup.Normal G'] (he : optParam (Subgroup.map (↑(MulEquiv.refl G)) G' = G') ⋯) : QuotientGroup.congr G' G' (MulEquiv.refl G) he = MulEquiv.refl (G ⧸ G')

@[simp]

theorem QuotientGroup.congr_symm {G : Type u} [Group G] {H : Type v} [Group H] (G' : Subgroup G) (H' : Subgroup H) [Subgroup.Normal G'] [Subgroup.Normal H'] (e : G ≃* H) (he : Subgroup.map (↑e) G' = H') : MulEquiv.symm (QuotientGroup.congr G' H' e he) = QuotientGroup.congr H' G' (MulEquiv.symm e) ⋯

def QuotientAddGroup.kerLift {G : Type u} [AddGroup G] {H : Type v} [AddGroup H] (φ : G →+ H) : G ⧸ AddMonoidHom.ker φ →+ H

The induced map from the quotient by the kernel to the codomain.

Equations

• QuotientAddGroup.kerLift φ = QuotientAddGroup.lift (AddMonoidHom.ker φ) φ ⋯

theorem QuotientAddGroup.kerLift.proof_1 {G : Type u_1} [AddGroup G] {H : Type u_2} [AddGroup H] (φ : G →+ H) : AddSubgroup.Normal (AddMonoidHom.ker φ)

theorem QuotientAddGroup.kerLift.proof_2 {G : Type u_1} [AddGroup G] {H : Type u_2} [AddGroup H] (φ : G →+ H) (_g : G) : _g ∈ AddMonoidHom.ker φ → φ _g = 0

def QuotientGroup.kerLift {G : Type u} [Group G] {H : Type v} [Group H] (φ : G →* H) : G ⧸ MonoidHom.ker φ →* H

The induced map from the quotient by the kernel to the codomain.

Equations

• QuotientGroup.kerLift φ = QuotientGroup.lift (MonoidHom.ker φ) φ ⋯

@[simp]

theorem QuotientAddGroup.kerLift_mk {G : Type u} [AddGroup G] {H : Type v} [AddGroup H] (φ : G →+ H) (g : G) : (QuotientAddGroup.kerLift φ) ↑g = φ g

@[simp]

theorem QuotientGroup.kerLift_mk {G : Type u} [Group G] {H : Type v} [Group H] (φ : G →* H) (g : G) : (QuotientGroup.kerLift φ) ↑g = φ g

@[simp]

theorem QuotientAddGroup.kerLift_mk' {G : Type u} [AddGroup G] {H : Type v} [AddGroup H] (φ : G →+ H) (g : G) : (QuotientAddGroup.kerLift φ) ↑g = φ g

@[simp]

theorem QuotientGroup.kerLift_mk' {G : Type u} [Group G] {H : Type v} [Group H] (φ : G →* H) (g : G) : (QuotientGroup.kerLift φ) ↑g = φ g

theorem QuotientAddGroup.kerLift_injective {G : Type u} [AddGroup G] {H : Type v} [AddGroup H] (φ : G →+ H) : Function.Injective ⇑(QuotientAddGroup.kerLift φ)

theorem QuotientGroup.kerLift_injective {G : Type u} [Group G] {H : Type v} [Group H] (φ : G →* H) : Function.Injective ⇑(QuotientGroup.kerLift φ)

def QuotientAddGroup.rangeKerLift {G : Type u} [AddGroup G] {H : Type v} [AddGroup H] (φ : G →+ H) : G ⧸ AddMonoidHom.ker φ →+ ↥(AddMonoidHom.range φ)

The induced map from the quotient by the kernel to the range.

Equations

• QuotientAddGroup.rangeKerLift φ = QuotientAddGroup.lift (AddMonoidHom.ker φ) (AddMonoidHom.rangeRestrict φ) ⋯

theorem QuotientAddGroup.rangeKerLift.proof_1 {G : Type u_1} [AddGroup G] {H : Type u_2} [AddGroup H] (φ : G →+ H) : AddSubgroup.Normal (AddMonoidHom.ker φ)

theorem QuotientAddGroup.rangeKerLift.proof_2 {G : Type u_1} [AddGroup G] {H : Type u_2} [AddGroup H] (φ : G →+ H) (g : G) (hg : g ∈ AddMonoidHom.ker φ) : (AddMonoidHom.rangeRestrict φ) g = 0

def QuotientGroup.rangeKerLift {G : Type u} [Group G] {H : Type v} [Group H] (φ : G →* H) : G ⧸ MonoidHom.ker φ →* ↥(MonoidHom.range φ)

The induced map from the quotient by the kernel to the range.

Equations

• QuotientGroup.rangeKerLift φ = QuotientGroup.lift (MonoidHom.ker φ) (MonoidHom.rangeRestrict φ) ⋯

theorem QuotientAddGroup.rangeKerLift_injective {G : Type u} [AddGroup G] {H : Type v} [AddGroup H] (φ : G →+ H) : Function.Injective ⇑(QuotientAddGroup.rangeKerLift φ)

theorem QuotientGroup.rangeKerLift_injective {G : Type u} [Group G] {H : Type v} [Group H] (φ : G →* H) : Function.Injective ⇑(QuotientGroup.rangeKerLift φ)

theorem QuotientAddGroup.rangeKerLift_surjective {G : Type u} [AddGroup G] {H : Type v} [AddGroup H] (φ : G →+ H) : Function.Surjective ⇑(QuotientAddGroup.rangeKerLift φ)

theorem QuotientGroup.rangeKerLift_surjective {G : Type u} [Group G] {H : Type v} [Group H] (φ : G →* H) : Function.Surjective ⇑(QuotientGroup.rangeKerLift φ)

theorem QuotientAddGroup.quotientKerEquivRange.proof_3 {G : Type u_1} [AddGroup G] {H : Type u_2} [AddGroup H] (φ : G →+ H) : Function.Injective ⇑(QuotientAddGroup.rangeKerLift φ) ∧ Function.Surjective ⇑(QuotientAddGroup.rangeKerLift φ)

noncomputable def QuotientAddGroup.quotientKerEquivRange {G : Type u} [AddGroup G] {H : Type v} [AddGroup H] (φ : G →+ H) : G ⧸ AddMonoidHom.ker φ ≃+ ↥(AddMonoidHom.range φ)

The first isomorphism theorem (a definition): the canonical isomorphism between G/(ker φ) to range φ.

Equations

• QuotientAddGroup.quotientKerEquivRange φ = AddEquiv.ofBijective (QuotientAddGroup.rangeKerLift φ) ⋯

theorem QuotientAddGroup.quotientKerEquivRange.proof_2 {G : Type u_2} [AddGroup G] {H : Type u_1} [AddGroup H] (φ : G →+ H) : AddHomClass (G ⧸ AddMonoidHom.ker φ →+ ↥(AddMonoidHom.range φ)) (G ⧸ AddMonoidHom.ker φ) ↥(AddMonoidHom.range φ)

theorem QuotientAddGroup.quotientKerEquivRange.proof_1 {G : Type u_1} [AddGroup G] {H : Type u_2} [AddGroup H] (φ : G →+ H) : AddSubgroup.Normal (AddMonoidHom.ker φ)

noncomputable def QuotientGroup.quotientKerEquivRange {G : Type u} [Group G] {H : Type v} [Group H] (φ : G →* H) : G ⧸ MonoidHom.ker φ ≃* ↥(MonoidHom.range φ)

Noether's first isomorphism theorem (a definition): the canonical isomorphism between G/(ker φ) to range φ.

Equations

• QuotientGroup.quotientKerEquivRange φ = MulEquiv.ofBijective (QuotientGroup.rangeKerLift φ) ⋯

theorem QuotientAddGroup.quotientKerEquivOfRightInverse.proof_3 {G : Type u_1} [AddGroup G] {H : Type u_2} [AddGroup H] (φ : G →+ H) (x : G ⧸ AddMonoidHom.ker φ) (y : G ⧸ AddMonoidHom.ker φ) : (QuotientAddGroup.kerLift φ).toFun (x + y) = (QuotientAddGroup.kerLift φ).toFun x + (QuotientAddGroup.kerLift φ).toFun y

theorem QuotientAddGroup.quotientKerEquivOfRightInverse.proof_2 {G : Type u_1} [AddGroup G] {H : Type u_2} [AddGroup H] (φ : G →+ H) (ψ : H → G) (hφ : Function.RightInverse ψ ⇑φ) (x : G ⧸ AddMonoidHom.ker φ) : (QuotientAddGroup.mk ∘ ψ) ((QuotientAddGroup.kerLift φ) x) = x

def QuotientAddGroup.quotientKerEquivOfRightInverse {G : Type u} [AddGroup G] {H : Type v} [AddGroup H] (φ : G →+ H) (ψ : H → G) (hφ : Function.RightInverse ψ ⇑φ) : G ⧸ AddMonoidHom.ker φ ≃+ H

The canonical isomorphism G/(ker φ) ≃+ H induced by a homomorphism φ : G →+ H with a right inverse ψ : H → G.

Equations

• One or more equations did not get rendered due to their size.

theorem QuotientAddGroup.quotientKerEquivOfRightInverse.proof_1 {G : Type u_1} [AddGroup G] {H : Type u_2} [AddGroup H] (φ : G →+ H) : AddSubgroup.Normal (AddMonoidHom.ker φ)

@[simp]

theorem QuotientAddGroup.quotientKerEquivOfRightInverse_apply {G : Type u} [AddGroup G] {H : Type v} [AddGroup H] (φ : G →+ H) (ψ : H → G) (hφ : Function.RightInverse ψ ⇑φ) (a : G ⧸ AddMonoidHom.ker φ) : (QuotientAddGroup.quotientKerEquivOfRightInverse φ ψ hφ) a = (QuotientAddGroup.kerLift φ) a

@[simp]

theorem QuotientAddGroup.quotientKerEquivOfRightInverse_symm_apply {G : Type u} [AddGroup G] {H : Type v} [AddGroup H] (φ : G →+ H) (ψ : H → G) (hφ : Function.RightInverse ψ ⇑φ) : ∀ (a : H), (AddEquiv.symm (QuotientAddGroup.quotientKerEquivOfRightInverse φ ψ hφ)) a = (QuotientAddGroup.mk ∘ ψ) a

@[simp]

theorem QuotientGroup.quotientKerEquivOfRightInverse_symm_apply {G : Type u} [Group G] {H : Type v} [Group H] (φ : G →* H) (ψ : H → G) (hφ : Function.RightInverse ψ ⇑φ) : ∀ (a : H), (MulEquiv.symm (QuotientGroup.quotientKerEquivOfRightInverse φ ψ hφ)) a = (QuotientGroup.mk ∘ ψ) a

@[simp]

theorem QuotientGroup.quotientKerEquivOfRightInverse_apply {G : Type u} [Group G] {H : Type v} [Group H] (φ : G →* H) (ψ : H → G) (hφ : Function.RightInverse ψ ⇑φ) (a : G ⧸ MonoidHom.ker φ) : (QuotientGroup.quotientKerEquivOfRightInverse φ ψ hφ) a = (QuotientGroup.kerLift φ) a

def QuotientGroup.quotientKerEquivOfRightInverse {G : Type u} [Group G] {H : Type v} [Group H] (φ : G →* H) (ψ : H → G) (hφ : Function.RightInverse ψ ⇑φ) : G ⧸ MonoidHom.ker φ ≃* H

The canonical isomorphism G/(ker φ) ≃* H induced by a homomorphism φ : G →* H with a right inverse ψ : H → G.

Equations

• One or more equations did not get rendered due to their size.

theorem QuotientAddGroup.quotientBot.proof_1 {G : Type u_1} [AddGroup G] (_x : G) : (AddMonoidHom.id G) (id _x) = (AddMonoidHom.id G) (id _x)

def QuotientAddGroup.quotientBot {G : Type u} [AddGroup G] : G ⧸ ⊥ ≃+ G

The canonical isomorphism G/⊥ ≃+ G.

Equations

• QuotientAddGroup.quotientBot = QuotientAddGroup.quotientKerEquivOfRightInverse (AddMonoidHom.id G) id ⋯

@[simp]

theorem QuotientGroup.quotientBot_apply {G : Type u} [Group G] (a : G ⧸ MonoidHom.ker (MonoidHom.id G)) : QuotientGroup.quotientBot a = (QuotientGroup.kerLift (MonoidHom.id G)) a

@[simp]

theorem QuotientAddGroup.quotientBot_symm_apply {G : Type u} [AddGroup G] : ∀ (a : G), (AddEquiv.symm QuotientAddGroup.quotientBot) a = ↑a

@[simp]

theorem QuotientAddGroup.quotientBot_apply {G : Type u} [AddGroup G] (a : G ⧸ AddMonoidHom.ker (AddMonoidHom.id G)) : QuotientAddGroup.quotientBot a = (QuotientAddGroup.kerLift (AddMonoidHom.id G)) a

@[simp]

theorem QuotientGroup.quotientBot_symm_apply {G : Type u} [Group G] : ∀ (a : G), (MulEquiv.symm QuotientGroup.quotientBot) a = ↑a

def QuotientGroup.quotientBot {G : Type u} [Group G] : G ⧸ ⊥ ≃* G

The canonical isomorphism G/⊥ ≃* G.

Equations

• QuotientGroup.quotientBot = QuotientGroup.quotientKerEquivOfRightInverse (MonoidHom.id G) id ⋯

theorem QuotientAddGroup.quotientKerEquivOfSurjective.proof_1 {G : Type u_1} [AddGroup G] {H : Type u_2} [AddGroup H] (φ : G →+ H) (hφ : Function.Surjective ⇑φ) : Function.HasRightInverse ⇑φ

theorem QuotientAddGroup.quotientKerEquivOfSurjective.proof_2 {G : Type u_1} [AddGroup G] {H : Type u_2} [AddGroup H] (φ : G →+ H) (hφ : Function.Surjective ⇑φ) : Function.RightInverse (Exists.choose ⋯) ⇑φ

noncomputable def QuotientAddGroup.quotientKerEquivOfSurjective {G : Type u} [AddGroup G] {H : Type v} [AddGroup H] (φ : G →+ H) (hφ : Function.Surjective ⇑φ) : G ⧸ AddMonoidHom.ker φ ≃+ H

The canonical isomorphism G/(ker φ) ≃+ H induced by a surjection φ : G →+ H. For a computable version, see QuotientAddGroup.quotientKerEquivOfRightInverse.

Equations

• QuotientAddGroup.quotientKerEquivOfSurjective φ hφ = QuotientAddGroup.quotientKerEquivOfRightInverse φ (Exists.choose ⋯) ⋯

noncomputable def QuotientGroup.quotientKerEquivOfSurjective {G : Type u} [Group G] {H : Type v} [Group H] (φ : G →* H) (hφ : Function.Surjective ⇑φ) : G ⧸ MonoidHom.ker φ ≃* H

The canonical isomorphism G/(ker φ) ≃* H induced by a surjection φ : G →* H.

For a computable version, see QuotientGroup.quotientKerEquivOfRightInverse.

Equations

• QuotientGroup.quotientKerEquivOfSurjective φ hφ = QuotientGroup.quotientKerEquivOfRightInverse φ (Exists.choose ⋯) ⋯

theorem QuotientAddGroup.quotientAddEquivOfEq.proof_1 {G : Type u_1} [AddGroup G] {M : AddSubgroup G} {N : AddSubgroup G} [AddSubgroup.Normal M] [AddSubgroup.Normal N] (h : M = N) (q : G ⧸ M) (r : G ⧸ M) : (AddSubgroup.quotientEquivOfEq h).toFun (q + r) = (AddSubgroup.quotientEquivOfEq h).toFun q + (AddSubgroup.quotientEquivOfEq h).toFun r

def QuotientAddGroup.quotientAddEquivOfEq {G : Type u} [AddGroup G] {M : AddSubgroup G} {N : AddSubgroup G} [AddSubgroup.Normal M] [AddSubgroup.Normal N] (h : M = N) : G ⧸ M ≃+ G ⧸ N

If two normal subgroups M and N of G are the same, their quotient groups are isomorphic.

Equations

• QuotientAddGroup.quotientAddEquivOfEq h = let __src := AddSubgroup.quotientEquivOfEq h; { toEquiv := __src, map_add' := ⋯ }

def QuotientGroup.quotientMulEquivOfEq {G : Type u} [Group G] {M : Subgroup G} {N : Subgroup G} [Subgroup.Normal M] [Subgroup.Normal N] (h : M = N) : G ⧸ M ≃* G ⧸ N

If two normal subgroups M and N of G are the same, their quotient groups are isomorphic.

Equations

• QuotientGroup.quotientMulEquivOfEq h = let __src := Subgroup.quotientEquivOfEq h; { toEquiv := __src, map_mul' := ⋯ }

@[simp]

theorem QuotientAddGroup.quotientAddEquivOfEq_mk {G : Type u} [AddGroup G] {M : AddSubgroup G} {N : AddSubgroup G} [AddSubgroup.Normal M] [AddSubgroup.Normal N] (h : M = N) (x : G) : (QuotientAddGroup.quotientAddEquivOfEq h) ↑x = ↑x

@[simp]

theorem QuotientGroup.quotientMulEquivOfEq_mk {G : Type u} [Group G] {M : Subgroup G} {N : Subgroup G} [Subgroup.Normal M] [Subgroup.Normal N] (h : M = N) (x : G) : (QuotientGroup.quotientMulEquivOfEq h) ↑x = ↑x

theorem QuotientAddGroup.quotientMapAddSubgroupOfOfLe.proof_1 {G : Type u_1} [AddGroup G] {A' : AddSubgroup G} {A : AddSubgroup G} {B' : AddSubgroup G} (h' : A' ≤ B') : AddSubgroup.comap (AddSubgroup.subtype A) A' ≤ AddSubgroup.comap (AddSubgroup.subtype A) B'

def QuotientAddGroup.quotientMapAddSubgroupOfOfLe {G : Type u} [AddGroup G] {A' : AddSubgroup G} {A : AddSubgroup G} {B' : AddSubgroup G} {B : AddSubgroup G} [_hAN : AddSubgroup.Normal (AddSubgroup.addSubgroupOf A' A)] [_hBN : AddSubgroup.Normal (AddSubgroup.addSubgroupOf B' B)] (h' : A' ≤ B') (h : A ≤ B) : ↥A ⧸ AddSubgroup.addSubgroupOf A' A →+ ↥B ⧸ AddSubgroup.addSubgroupOf B' B

Let A', A, B', B be subgroups of G. If A' ≤ B' and A ≤ B, then there is a map A / (A' ⊓ A) →+ B / (B' ⊓ B) induced by the inclusions.

Equations

• QuotientAddGroup.quotientMapAddSubgroupOfOfLe h' h = QuotientAddGroup.map (AddSubgroup.addSubgroupOf A' A) (AddSubgroup.addSubgroupOf B' B) (AddSubgroup.inclusion h) ⋯

def QuotientGroup.quotientMapSubgroupOfOfLe {G : Type u} [Group G] {A' : Subgroup G} {A : Subgroup G} {B' : Subgroup G} {B : Subgroup G} [_hAN : Subgroup.Normal (Subgroup.subgroupOf A' A)] [_hBN : Subgroup.Normal (Subgroup.subgroupOf B' B)] (h' : A' ≤ B') (h : A ≤ B) : ↥A ⧸ Subgroup.subgroupOf A' A →* ↥B ⧸ Subgroup.subgroupOf B' B

Let A', A, B', B be subgroups of G. If A' ≤ B' and A ≤ B, then there is a map A / (A' ⊓ A) →* B / (B' ⊓ B) induced by the inclusions.

Equations

• QuotientGroup.quotientMapSubgroupOfOfLe h' h = QuotientGroup.map (Subgroup.subgroupOf A' A) (Subgroup.subgroupOf B' B) (Subgroup.inclusion h) ⋯

@[simp]

theorem QuotientAddGroup.quotientMapAddSubgroupOfOfLe_mk {G : Type u} [AddGroup G] {A' : AddSubgroup G} {A : AddSubgroup G} {B' : AddSubgroup G} {B : AddSubgroup G} [_hAN : AddSubgroup.Normal (AddSubgroup.addSubgroupOf A' A)] [_hBN : AddSubgroup.Normal (AddSubgroup.addSubgroupOf B' B)] (h' : A' ≤ B') (h : A ≤ B) (x : ↥A) : (QuotientAddGroup.quotientMapAddSubgroupOfOfLe h' h) ↑x = ↑((AddSubgroup.inclusion h) x)

@[simp]

theorem QuotientGroup.quotientMapSubgroupOfOfLe_mk {G : Type u} [Group G] {A' : Subgroup G} {A : Subgroup G} {B' : Subgroup G} {B : Subgroup G} [_hAN : Subgroup.Normal (Subgroup.subgroupOf A' A)] [_hBN : Subgroup.Normal (Subgroup.subgroupOf B' B)] (h' : A' ≤ B') (h : A ≤ B) (x : ↥A) : (QuotientGroup.quotientMapSubgroupOfOfLe h' h) ↑x = ↑((Subgroup.inclusion h) x)

theorem QuotientAddGroup.equivQuotientAddSubgroupOfOfEq.proof_2 {G : Type u_1} [AddGroup G] {A : AddSubgroup G} {B : AddSubgroup G} (h : A = B) : A ≤ B

theorem QuotientAddGroup.equivQuotientAddSubgroupOfOfEq.proof_3 {G : Type u_1} [AddGroup G] {A' : AddSubgroup G} {B' : AddSubgroup G} (h' : A' = B') : B' ≤ A'

theorem QuotientAddGroup.equivQuotientAddSubgroupOfOfEq.proof_1 {G : Type u_1} [AddGroup G] {A' : AddSubgroup G} {B' : AddSubgroup G} (h' : A' = B') : A' ≤ B'

def QuotientAddGroup.equivQuotientAddSubgroupOfOfEq {G : Type u} [AddGroup G] {A' : AddSubgroup G} {A : AddSubgroup G} {B' : AddSubgroup G} {B : AddSubgroup G} [hAN : AddSubgroup.Normal (AddSubgroup.addSubgroupOf A' A)] [hBN : AddSubgroup.Normal (AddSubgroup.addSubgroupOf B' B)] (h' : A' = B') (h : A = B) : ↥A ⧸ AddSubgroup.addSubgroupOf A' A ≃+ ↥B ⧸ AddSubgroup.addSubgroupOf B' B

Let A', A, B', B be subgroups of G. If A' = B' and A = B, then the quotients A / (A' ⊓ A) and B / (B' ⊓ B) are isomorphic. Applying this equiv is nicer than rewriting along the equalities, since the type of (A'.addSubgroupOf A : AddSubgroup A) depends on on A.

Equations

• QuotientAddGroup.equivQuotientAddSubgroupOfOfEq h' h = AddMonoidHom.toAddEquiv (QuotientAddGroup.quotientMapAddSubgroupOfOfLe ⋯ ⋯) (QuotientAddGroup.quotientMapAddSubgroupOfOfLe ⋯ ⋯) ⋯ ⋯

theorem QuotientAddGroup.equivQuotientAddSubgroupOfOfEq.proof_6 {G : Type u_1} [AddGroup G] {A' : AddSubgroup G} {A : AddSubgroup G} {B' : AddSubgroup G} {B : AddSubgroup G} [hAN : AddSubgroup.Normal (AddSubgroup.addSubgroupOf A' A)] [hBN : AddSubgroup.Normal (AddSubgroup.addSubgroupOf B' B)] (h' : A' = B') (h : A = B) : AddMonoidHom.comp (QuotientAddGroup.quotientMapAddSubgroupOfOfLe ⋯ ⋯) (QuotientAddGroup.quotientMapAddSubgroupOfOfLe ⋯ ⋯) = AddMonoidHom.id (↥B ⧸ AddSubgroup.addSubgroupOf B' B)

theorem QuotientAddGroup.equivQuotientAddSubgroupOfOfEq.proof_5 {G : Type u_1} [AddGroup G] {A' : AddSubgroup G} {A : AddSubgroup G} {B' : AddSubgroup G} {B : AddSubgroup G} [hAN : AddSubgroup.Normal (AddSubgroup.addSubgroupOf A' A)] [hBN : AddSubgroup.Normal (AddSubgroup.addSubgroupOf B' B)] (h' : A' = B') (h : A = B) : AddMonoidHom.comp (QuotientAddGroup.quotientMapAddSubgroupOfOfLe ⋯ ⋯) (QuotientAddGroup.quotientMapAddSubgroupOfOfLe ⋯ ⋯) = AddMonoidHom.id (↥A ⧸ AddSubgroup.addSubgroupOf A' A)

theorem QuotientAddGroup.equivQuotientAddSubgroupOfOfEq.proof_4 {G : Type u_1} [AddGroup G] {A : AddSubgroup G} {B : AddSubgroup G} (h : A = B) : B ≤ A

def QuotientGroup.equivQuotientSubgroupOfOfEq {G : Type u} [Group G] {A' : Subgroup G} {A : Subgroup G} {B' : Subgroup G} {B : Subgroup G} [hAN : Subgroup.Normal (Subgroup.subgroupOf A' A)] [hBN : Subgroup.Normal (Subgroup.subgroupOf B' B)] (h' : A' = B') (h : A = B) : ↥A ⧸ Subgroup.subgroupOf A' A ≃* ↥B ⧸ Subgroup.subgroupOf B' B

Let A', A, B', B be subgroups of G. If A' = B' and A = B, then the quotients A / (A' ⊓ A) and B / (B' ⊓ B) are isomorphic.

Applying this equiv is nicer than rewriting along the equalities, since the type of (A'.subgroupOf A : Subgroup A) depends on A.

Equations

• QuotientGroup.equivQuotientSubgroupOfOfEq h' h = MonoidHom.toMulEquiv (QuotientGroup.quotientMapSubgroupOfOfLe ⋯ ⋯) (QuotientGroup.quotientMapSubgroupOfOfLe ⋯ ⋯) ⋯ ⋯

theorem QuotientAddGroup.homQuotientZSMulOfHom.proof_1 {A : Type u_1} [AddCommGroup A] (n : ℤ) : AddSubgroup.Normal (AddMonoidHom.range (zsmulAddGroupHom n))

abbrev QuotientAddGroup.homQuotientZSMulOfHom.match_1 {A : Type u_1} [AddCommGroup A] (n : ℤ) (g : A) (motive : g ∈ AddMonoidHom.range (zsmulAddGroupHom n) → Prop) : ∀ (x : g ∈ AddMonoidHom.range (zsmulAddGroupHom n)), (∀ (h : A) (hg : n • h = g), motive ⋯) → motive x

Equations

• ⋯ = ⋯

theorem QuotientAddGroup.homQuotientZSMulOfHom.proof_2 {B : Type u_1} [AddCommGroup B] (n : ℤ) : AddSubgroup.Normal (AddMonoidHom.range (zsmulAddGroupHom n))

def QuotientAddGroup.homQuotientZSMulOfHom {A : Type u} {B : Type u} [AddCommGroup A] [AddCommGroup B] (f : A →+ B) (n : ℤ) : A ⧸ AddMonoidHom.range (zsmulAddGroupHom n) →+ B ⧸ AddMonoidHom.range (zsmulAddGroupHom n)

The map of quotients by multiples of an integer induced by an additive group homomorphism.

Equations

• One or more equations did not get rendered due to their size.

theorem QuotientAddGroup.homQuotientZSMulOfHom.proof_3 {A : Type u_1} {B : Type u_1} [AddCommGroup A] [AddCommGroup B] (f : A →+ B) (n : ℤ) (g : A) : g ∈ AddMonoidHom.range (zsmulAddGroupHom n) → g ∈ AddMonoidHom.ker (AddMonoidHom.comp (QuotientAddGroup.mk' (AddMonoidHom.range (zsmulAddGroupHom n))) f)

def QuotientGroup.homQuotientZPowOfHom {A : Type u} {B : Type u} [CommGroup A] [CommGroup B] (f : A →* B) (n : ℤ) : A ⧸ MonoidHom.range (zpowGroupHom n) →* B ⧸ MonoidHom.range (zpowGroupHom n)

The map of quotients by powers of an integer induced by a group homomorphism.

Equations

• QuotientGroup.homQuotientZPowOfHom f n = QuotientGroup.lift (MonoidHom.range (zpowGroupHom n)) (MonoidHom.comp (QuotientGroup.mk' (MonoidHom.range (zpowGroupHom n))) f) ⋯

@[simp]

theorem QuotientAddGroup.homQuotientZSMulOfHom_id {A : Type u} [AddCommGroup A] (n : ℤ) : QuotientAddGroup.homQuotientZSMulOfHom (AddMonoidHom.id A) n = AddMonoidHom.id (A ⧸ AddMonoidHom.range (zsmulAddGroupHom n))

@[simp]

theorem QuotientGroup.homQuotientZPowOfHom_id {A : Type u} [CommGroup A] (n : ℤ) : QuotientGroup.homQuotientZPowOfHom (MonoidHom.id A) n = MonoidHom.id (A ⧸ MonoidHom.range (zpowGroupHom n))

@[simp]

theorem QuotientAddGroup.homQuotientZSMulOfHom_comp {A : Type u} {B : Type u} [AddCommGroup A] [AddCommGroup B] (f : A →+ B) (g : B →+ A) (n : ℤ) : QuotientAddGroup.homQuotientZSMulOfHom (AddMonoidHom.comp f g) n = AddMonoidHom.comp (QuotientAddGroup.homQuotientZSMulOfHom f n) (QuotientAddGroup.homQuotientZSMulOfHom g n)

@[simp]

theorem QuotientGroup.homQuotientZPowOfHom_comp {A : Type u} {B : Type u} [CommGroup A] [CommGroup B] (f : A →* B) (g : B →* A) (n : ℤ) : QuotientGroup.homQuotientZPowOfHom (MonoidHom.comp f g) n = MonoidHom.comp (QuotientGroup.homQuotientZPowOfHom f n) (QuotientGroup.homQuotientZPowOfHom g n)

@[simp]

theorem QuotientAddGroup.homQuotientZSMulOfHom_comp_of_rightInverse {A : Type u} {B : Type u} [AddCommGroup A] [AddCommGroup B] (f : A →+ B) (g : B →+ A) (n : ℤ) (i : Function.RightInverse ⇑g ⇑f) : AddMonoidHom.comp (QuotientAddGroup.homQuotientZSMulOfHom f n) (QuotientAddGroup.homQuotientZSMulOfHom g n) = AddMonoidHom.id (B ⧸ AddMonoidHom.range (zsmulAddGroupHom n))

@[simp]

theorem QuotientGroup.homQuotientZPowOfHom_comp_of_rightInverse {A : Type u} {B : Type u} [CommGroup A] [CommGroup B] (f : A →* B) (g : B →* A) (n : ℤ) (i : Function.RightInverse ⇑g ⇑f) : MonoidHom.comp (QuotientGroup.homQuotientZPowOfHom f n) (QuotientGroup.homQuotientZPowOfHom g n) = MonoidHom.id (B ⧸ MonoidHom.range (zpowGroupHom n))

def QuotientAddGroup.equivQuotientZSMulOfEquiv {A : Type u} {B : Type u} [AddCommGroup A] [AddCommGroup B] (e : A ≃+ B) (n : ℤ) : A ⧸ AddMonoidHom.range (zsmulAddGroupHom n) ≃+ B ⧸ AddMonoidHom.range (zsmulAddGroupHom n)

The equivalence of quotients by multiples of an integer induced by an additive group isomorphism.

Equations

• QuotientAddGroup.equivQuotientZSMulOfEquiv e n = AddMonoidHom.toAddEquiv (QuotientAddGroup.homQuotientZSMulOfHom (↑e) n) (QuotientAddGroup.homQuotientZSMulOfHom (↑(AddEquiv.symm e)) n) ⋯ ⋯

theorem QuotientAddGroup.equivQuotientZSMulOfEquiv.proof_2 {B : Type u_1} [AddCommGroup B] (n : ℤ) : AddSubgroup.Normal (AddMonoidHom.range (zsmulAddGroupHom n))

theorem QuotientAddGroup.equivQuotientZSMulOfEquiv.proof_1 {A : Type u_1} [AddCommGroup A] (n : ℤ) : AddSubgroup.Normal (AddMonoidHom.range (zsmulAddGroupHom n))

theorem QuotientAddGroup.equivQuotientZSMulOfEquiv.proof_4 {A : Type u_1} {B : Type u_1} [AddCommGroup A] [AddCommGroup B] : AddMonoidHomClass (B ≃+ A) B A

theorem QuotientAddGroup.equivQuotientZSMulOfEquiv.proof_5 {A : Type u_1} {B : Type u_1} [AddCommGroup A] [AddCommGroup B] (e : A ≃+ B) (n : ℤ) : AddMonoidHom.comp (QuotientAddGroup.homQuotientZSMulOfHom (↑(AddEquiv.symm e)) n) (QuotientAddGroup.homQuotientZSMulOfHom (↑e) n) = AddMonoidHom.id (A ⧸ AddMonoidHom.range (zsmulAddGroupHom n))

theorem QuotientAddGroup.equivQuotientZSMulOfEquiv.proof_3 {A : Type u_1} {B : Type u_1} [AddCommGroup A] [AddCommGroup B] : AddMonoidHomClass (A ≃+ B) A B

theorem QuotientAddGroup.equivQuotientZSMulOfEquiv.proof_6 {A : Type u_1} {B : Type u_1} [AddCommGroup A] [AddCommGroup B] (e : A ≃+ B) (n : ℤ) : AddMonoidHom.comp (QuotientAddGroup.homQuotientZSMulOfHom (↑e) n) (QuotientAddGroup.homQuotientZSMulOfHom (↑(AddEquiv.symm e)) n) = AddMonoidHom.id (B ⧸ AddMonoidHom.range (zsmulAddGroupHom n))

def QuotientGroup.equivQuotientZPowOfEquiv {A : Type u} {B : Type u} [CommGroup A] [CommGroup B] (e : A ≃* B) (n : ℤ) : A ⧸ MonoidHom.range (zpowGroupHom n) ≃* B ⧸ MonoidHom.range (zpowGroupHom n)

The equivalence of quotients by powers of an integer induced by a group isomorphism.

Equations

• QuotientGroup.equivQuotientZPowOfEquiv e n = MonoidHom.toMulEquiv (QuotientGroup.homQuotientZPowOfHom (↑e) n) (QuotientGroup.homQuotientZPowOfHom (↑(MulEquiv.symm e)) n) ⋯ ⋯

@[simp]

theorem QuotientAddGroup.equivQuotientZSMulOfEquiv_refl {A : Type u} [AddCommGroup A] (n : ℤ) : AddEquiv.refl (A ⧸ AddMonoidHom.range (zsmulAddGroupHom n)) = QuotientAddGroup.equivQuotientZSMulOfEquiv (AddEquiv.refl A) n

@[simp]

theorem QuotientGroup.equivQuotientZPowOfEquiv_refl {A : Type u} [CommGroup A] (n : ℤ) : MulEquiv.refl (A ⧸ MonoidHom.range (zpowGroupHom n)) = QuotientGroup.equivQuotientZPowOfEquiv (MulEquiv.refl A) n

@[simp]

theorem QuotientAddGroup.equivQuotientZSMulOfEquiv_symm {A : Type u} {B : Type u} [AddCommGroup A] [AddCommGroup B] (e : A ≃+ B) (n : ℤ) : AddEquiv.symm (QuotientAddGroup.equivQuotientZSMulOfEquiv e n) = QuotientAddGroup.equivQuotientZSMulOfEquiv (AddEquiv.symm e) n

@[simp]

theorem QuotientGroup.equivQuotientZPowOfEquiv_symm {A : Type u} {B : Type u} [CommGroup A] [CommGroup B] (e : A ≃* B) (n : ℤ) : MulEquiv.symm (QuotientGroup.equivQuotientZPowOfEquiv e n) = QuotientGroup.equivQuotientZPowOfEquiv (MulEquiv.symm e) n

@[simp]

theorem QuotientAddGroup.equivQuotientZSMulOfEquiv_trans {A : Type u} {B : Type u} {C : Type u} [AddCommGroup A] [AddCommGroup B] [AddCommGroup C] (e : A ≃+ B) (d : B ≃+ C) (n : ℤ) : AddEquiv.trans (QuotientAddGroup.equivQuotientZSMulOfEquiv e n) (QuotientAddGroup.equivQuotientZSMulOfEquiv d n) = QuotientAddGroup.equivQuotientZSMulOfEquiv (AddEquiv.trans e d) n

@[simp]

theorem QuotientGroup.equivQuotientZPowOfEquiv_trans {A : Type u} {B : Type u} {C : Type u} [CommGroup A] [CommGroup B] [CommGroup C] (e : A ≃* B) (d : B ≃* C) (n : ℤ) : MulEquiv.trans (QuotientGroup.equivQuotientZPowOfEquiv e n) (QuotientGroup.equivQuotientZPowOfEquiv d n) = QuotientGroup.equivQuotientZPowOfEquiv (MulEquiv.trans e d) n

theorem QuotientAddGroup.quotientInfEquivSumNormalQuotient.proof_2 {G : Type u_1} [AddGroup G] (H : AddSubgroup G) (N : AddSubgroup G) : H ≤ H ⊔ N

theorem QuotientAddGroup.quotientInfEquivSumNormalQuotient.proof_1 {G : Type u_1} [AddGroup G] (H : AddSubgroup G) (N : AddSubgroup G) [AddSubgroup.Normal N] : AddSubgroup.Normal (AddSubgroup.addSubgroupOf N (H ⊔ N))

theorem QuotientAddGroup.quotientInfEquivSumNormalQuotient.proof_3 {G : Type u_1} [AddGroup G] (H : AddSubgroup G) (N : AddSubgroup G) [AddSubgroup.Normal N] (x : ↥(H ⊔ N) ⧸ AddSubgroup.addSubgroupOf N (H ⊔ N)) : ∃ (a : ↥H), (AddMonoidHom.comp (QuotientAddGroup.mk' (AddSubgroup.addSubgroupOf N (H ⊔ N))) (AddSubgroup.inclusion ⋯)) a = x

noncomputable def QuotientAddGroup.quotientInfEquivSumNormalQuotient {G : Type u} [AddGroup G] (H : AddSubgroup G) (N : AddSubgroup G) [AddSubgroup.Normal N] : ↥H ⧸ AddSubgroup.addSubgroupOf N H ≃+ ↥(H ⊔ N) ⧸ AddSubgroup.addSubgroupOf N (H ⊔ N)

The second isomorphism theorem: given two subgroups H and N of a group G, where N is normal, defines an isomorphism between H/(H ∩ N) and (H + N)/N

Equations

• One or more equations did not get rendered due to their size.

theorem QuotientAddGroup.quotientInfEquivSumNormalQuotient.proof_5 {G : Type u_1} [AddGroup G] (H : AddSubgroup G) (N : AddSubgroup G) [AddSubgroup.Normal N] : AddSubgroup.addSubgroupOf N H = AddSubgroup.comap (AddSubgroup.inclusion ⋯) (AddMonoidHom.ker (QuotientAddGroup.mk' (AddSubgroup.addSubgroupOf N (H ⊔ N))))

theorem QuotientAddGroup.quotientInfEquivSumNormalQuotient.proof_4 {G : Type u_1} [AddGroup G] (H : AddSubgroup G) (N : AddSubgroup G) [AddSubgroup.Normal N] : AddSubgroup.Normal (AddMonoidHom.ker (AddMonoidHom.comp (QuotientAddGroup.mk' (AddSubgroup.addSubgroupOf N (H ⊔ N))) (AddSubgroup.inclusion ⋯)))

noncomputable def QuotientGroup.quotientInfEquivProdNormalQuotient {G : Type u} [Group G] (H : Subgroup G) (N : Subgroup G) [Subgroup.Normal N] : ↥H ⧸ Subgroup.subgroupOf N H ≃* ↥(H ⊔ N) ⧸ Subgroup.subgroupOf N (H ⊔ N)

Noether's second isomorphism theorem: given two subgroups H and N of a group G, where N is normal, defines an isomorphism between H/(H ∩ N) and (HN)/N.

Equations

• One or more equations did not get rendered due to their size.

instance QuotientAddGroup.map_normal {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] (M : AddSubgroup G) [nM : AddSubgroup.Normal M] : AddSubgroup.Normal (AddSubgroup.map (QuotientAddGroup.mk' N) M)

Equations

• ⋯ = ⋯

instance QuotientGroup.map_normal {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] (M : Subgroup G) [nM : Subgroup.Normal M] : Subgroup.Normal (Subgroup.map (QuotientGroup.mk' N) M)

Equations

• ⋯ = ⋯

def QuotientAddGroup.quotientQuotientEquivQuotientAux {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] (M : AddSubgroup G) [nM : AddSubgroup.Normal M] (h : N ≤ M) : (G ⧸ N) ⧸ AddSubgroup.map (QuotientAddGroup.mk' N) M →+ G ⧸ M

The map from the third isomorphism theorem for additive groups: (A / N) / (M / N) → A / M.

Equations

• QuotientAddGroup.quotientQuotientEquivQuotientAux N M h = QuotientAddGroup.lift (AddSubgroup.map (QuotientAddGroup.mk' N) M) (QuotientAddGroup.map N M (AddMonoidHom.id G) h) ⋯

theorem QuotientAddGroup.quotientQuotientEquivQuotientAux.proof_1 {G : Type u_1} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] (M : AddSubgroup G) [nM : AddSubgroup.Normal M] (h : N ≤ M) : ∀ ⦃x : G ⧸ N⦄, x ∈ AddSubgroup.map (QuotientAddGroup.mk' N) M → x ∈ AddMonoidHom.ker (QuotientAddGroup.map N M (AddMonoidHom.id G) h)

def QuotientGroup.quotientQuotientEquivQuotientAux {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] (M : Subgroup G) [nM : Subgroup.Normal M] (h : N ≤ M) : (G ⧸ N) ⧸ Subgroup.map (QuotientGroup.mk' N) M →* G ⧸ M

The map from the third isomorphism theorem for groups: (G / N) / (M / N) → G / M.

Equations

• QuotientGroup.quotientQuotientEquivQuotientAux N M h = QuotientGroup.lift (Subgroup.map (QuotientGroup.mk' N) M) (QuotientGroup.map N M (MonoidHom.id G) h) ⋯

@[simp]

theorem QuotientAddGroup.quotientQuotientEquivQuotientAux_mk {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] (M : AddSubgroup G) [nM : AddSubgroup.Normal M] (h : N ≤ M) (x : G ⧸ N) : (QuotientAddGroup.quotientQuotientEquivQuotientAux N M h) ↑x = (QuotientAddGroup.map N M (AddMonoidHom.id G) h) x

@[simp]

theorem QuotientGroup.quotientQuotientEquivQuotientAux_mk {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] (M : Subgroup G) [nM : Subgroup.Normal M] (h : N ≤ M) (x : G ⧸ N) : (QuotientGroup.quotientQuotientEquivQuotientAux N M h) ↑x = (QuotientGroup.map N M (MonoidHom.id G) h) x

theorem QuotientAddGroup.quotientQuotientEquivQuotientAux_mk_mk {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] (M : AddSubgroup G) [nM : AddSubgroup.Normal M] (h : N ≤ M) (x : G) : (QuotientAddGroup.quotientQuotientEquivQuotientAux N M h) ↑↑x = ↑x

theorem QuotientGroup.quotientQuotientEquivQuotientAux_mk_mk {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] (M : Subgroup G) [nM : Subgroup.Normal M] (h : N ≤ M) (x : G) : (QuotientGroup.quotientQuotientEquivQuotientAux N M h) ↑↑x = ↑x

def QuotientAddGroup.quotientQuotientEquivQuotient {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] (M : AddSubgroup G) [nM : AddSubgroup.Normal M] (h : N ≤ M) : (G ⧸ N) ⧸ AddSubgroup.map (QuotientAddGroup.mk' N) M ≃+ G ⧸ M

Noether's third isomorphism theorem for additive groups: (A / N) / (M / N) ≃+ A / M.

Equations

• One or more equations did not get rendered due to their size.

theorem QuotientAddGroup.quotientQuotientEquivQuotient.proof_3 {G : Type u_1} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] (M : AddSubgroup G) [nM : AddSubgroup.Normal M] (h : N ≤ M) : AddMonoidHom.comp (QuotientAddGroup.quotientQuotientEquivQuotientAux N M h) (QuotientAddGroup.map M (AddSubgroup.map (QuotientAddGroup.mk' N) M) (QuotientAddGroup.mk' N) ⋯) = AddMonoidHom.id (G ⧸ M)

theorem QuotientAddGroup.quotientQuotientEquivQuotient.proof_1 {G : Type u_1} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] (M : AddSubgroup G) : M ≤ AddSubgroup.comap (QuotientAddGroup.mk' N) (AddSubgroup.map (QuotientAddGroup.mk' N) M)

theorem QuotientAddGroup.quotientQuotientEquivQuotient.proof_2 {G : Type u_1} [AddGroup G] (N : AddSubgroup G) [nN : AddSubgroup.Normal N] (M : AddSubgroup G) [nM : AddSubgroup.Normal M] (h : N ≤ M) : AddMonoidHom.comp (QuotientAddGroup.map M (AddSubgroup.map (QuotientAddGroup.mk' N) M) (QuotientAddGroup.mk' N) ⋯) (QuotientAddGroup.quotientQuotientEquivQuotientAux N M h) = AddMonoidHom.id ((G ⧸ N) ⧸ AddSubgroup.map (QuotientAddGroup.mk' N) M)

def QuotientGroup.quotientQuotientEquivQuotient {G : Type u} [Group G] (N : Subgroup G) [nN : Subgroup.Normal N] (M : Subgroup G) [nM : Subgroup.Normal M] (h : N ≤ M) : (G ⧸ N) ⧸ Subgroup.map (QuotientGroup.mk' N) M ≃* G ⧸ M

Noether's third isomorphism theorem for groups: (G / N) / (M / N) ≃* G / M.

Equations

• One or more equations did not get rendered due to their size.

theorem QuotientAddGroup.subsingleton_quotient_top {G : Type u} [AddGroup G] : Subsingleton (G ⧸ ⊤)

theorem QuotientGroup.subsingleton_quotient_top {G : Type u} [Group G] : Subsingleton (G ⧸ ⊤)

theorem QuotientAddGroup.addSubgroup_eq_top_of_subsingleton {G : Type u} [AddGroup G] (H : AddSubgroup G) (h : Subsingleton (G ⧸ H)) : H = ⊤

If the quotient by an additive subgroup gives a singleton then the additive subgroup is the whole additive group.

theorem QuotientGroup.subgroup_eq_top_of_subsingleton {G : Type u} [Group G] (H : Subgroup G) (h : Subsingleton (G ⧸ H)) : H = ⊤

If the quotient by a subgroup gives a singleton then the subgroup is the whole group.

theorem QuotientAddGroup.comap_comap_center {G : Type u} [AddGroup G] {H₁ : AddSubgroup G} [AddSubgroup.Normal H₁] {H₂ : AddSubgroup (G ⧸ H₁)} [AddSubgroup.Normal H₂] : AddSubgroup.comap (QuotientAddGroup.mk' H₁) (AddSubgroup.comap (QuotientAddGroup.mk' H₂) (AddSubgroup.center ((G ⧸ H₁) ⧸ H₂))) = AddSubgroup.comap (QuotientAddGroup.mk' (AddSubgroup.comap (QuotientAddGroup.mk' H₁) H₂)) (AddSubgroup.center (G ⧸ AddSubgroup.comap (QuotientAddGroup.mk' H₁) H₂))

theorem QuotientGroup.comap_comap_center {G : Type u} [Group G] {H₁ : Subgroup G} [Subgroup.Normal H₁] {H₂ : Subgroup (G ⧸ H₁)} [Subgroup.Normal H₂] : Subgroup.comap (QuotientGroup.mk' H₁) (Subgroup.comap (QuotientGroup.mk' H₂) (Subgroup.center ((G ⧸ H₁) ⧸ H₂))) = Subgroup.comap (QuotientGroup.mk' (Subgroup.comap (QuotientGroup.mk' H₁) H₂)) (Subgroup.center (G ⧸ Subgroup.comap (QuotientGroup.mk' H₁) H₂))

theorem AddGroup.fintypeOfKerLeRange.proof_2 {F : Type u_1} {G : Type u_1} {H : Type u_1} [AddGroup F] [AddGroup G] [AddGroup H] (f : F →+ G) (g : G →+ H) (h : AddMonoidHom.ker g ≤ AddMonoidHom.range f) : Function.Injective ⇑(AddSubgroup.inclusion h)

theorem AddGroup.fintypeOfKerLeRange.proof_1 {G : Type u_1} {H : Type u_1} [AddGroup G] [AddGroup H] (g : G →+ H) : AddSubgroup.Normal (AddMonoidHom.ker g)

noncomputable def AddGroup.fintypeOfKerLeRange {F : Type u} {G : Type u} {H : Type u} [AddGroup F] [AddGroup G] [AddGroup H] [Fintype F] [Fintype H] (f : F →+ G) (g : G →+ H) (h : AddMonoidHom.ker g ≤ AddMonoidHom.range f) : Fintype G

If F and H are finite such that ker(G →+ H) ≤ im(F →+ G), then G is finite.

Equations

• AddGroup.fintypeOfKerLeRange f g h = Fintype.ofEquiv ((G ⧸ AddMonoidHom.ker g) × ↥(AddMonoidHom.ker g)) AddSubgroup.addGroupEquivQuotientProdAddSubgroup.symm

noncomputable def Group.fintypeOfKerLeRange {F : Type u} {G : Type u} {H : Type u} [Group F] [Group G] [Group H] [Fintype F] [Fintype H] (f : F →* G) (g : G →* H) (h : MonoidHom.ker g ≤ MonoidHom.range f) : Fintype G

If F and H are finite such that ker(G →* H) ≤ im(F →* G), then G is finite.

Equations

• Group.fintypeOfKerLeRange f g h = Fintype.ofEquiv ((G ⧸ MonoidHom.ker g) × ↥(MonoidHom.ker g)) Subgroup.groupEquivQuotientProdSubgroup.symm

theorem AddGroup.fintypeOfKerEqRange.proof_1 {F : Type u_1} {G : Type u_1} {H : Type u_1} [AddGroup F] [AddGroup G] [AddGroup H] (f : F →+ G) (g : G →+ H) (h : AddMonoidHom.ker g = AddMonoidHom.range f) : AddMonoidHom.ker g ≤ AddMonoidHom.range f

noncomputable def AddGroup.fintypeOfKerEqRange {F : Type u} {G : Type u} {H : Type u} [AddGroup F] [AddGroup G] [AddGroup H] [Fintype F] [Fintype H] (f : F →+ G) (g : G →+ H) (h : AddMonoidHom.ker g = AddMonoidHom.range f) : Fintype G

If F and H are finite such that ker(G →+ H) = im(F →+ G), then G is finite.

Equations

• AddGroup.fintypeOfKerEqRange f g h = AddGroup.fintypeOfKerLeRange f g ⋯

noncomputable def Group.fintypeOfKerEqRange {F : Type u} {G : Type u} {H : Type u} [Group F] [Group G] [Group H] [Fintype F] [Fintype H] (f : F →* G) (g : G →* H) (h : MonoidHom.ker g = MonoidHom.range f) : Fintype G

If F and H are finite such that ker(G →* H) = im(F →* G), then G is finite.

Equations

• Group.fintypeOfKerEqRange f g h = Group.fintypeOfKerLeRange f g ⋯

theorem AddGroup.fintypeOfKerOfCodom.proof_2 {G : Type u_1} {H : Type u_1} [AddGroup G] [AddGroup H] (g : G →+ H) (x : G) (hx : x ∈ AddMonoidHom.ker g) : ∃ (y : ↥(AddMonoidHom.ker g)), (AddMonoidHom.comp (AddEquiv.toAddMonoidHom AddSubgroup.topEquiv) (AddSubgroup.inclusion ⋯)) y = x

noncomputable def AddGroup.fintypeOfKerOfCodom {G : Type u} {H : Type u} [AddGroup G] [AddGroup H] [Fintype H] (g : G →+ H) [Fintype ↥(AddMonoidHom.ker g)] : Fintype G

If ker(G →+ H) and H are finite, then G is finite.

Equations

• AddGroup.fintypeOfKerOfCodom g = AddGroup.fintypeOfKerLeRange (AddMonoidHom.comp (AddEquiv.toAddMonoidHom AddSubgroup.topEquiv) (AddSubgroup.inclusion ⋯)) g ⋯

theorem AddGroup.fintypeOfKerOfCodom.proof_1 {G : Type u_1} {H : Type u_1} [AddGroup G] [AddGroup H] (g : G →+ H) : AddMonoidHom.ker g ≤ ⊤

noncomputable def Group.fintypeOfKerOfCodom {G : Type u} {H : Type u} [Group G] [Group H] [Fintype H] (g : G →* H) [Fintype ↥(MonoidHom.ker g)] : Fintype G

If ker(G →* H) and H are finite, then G is finite.

Equations

• Group.fintypeOfKerOfCodom g = Group.fintypeOfKerLeRange (MonoidHom.comp (MulEquiv.toMonoidHom Subgroup.topEquiv) (Subgroup.inclusion ⋯)) g ⋯

theorem AddGroup.fintypeOfDomOfCoker.proof_1 {F : Type u_1} {G : Type u_1} [AddGroup F] [AddGroup G] (f : F →+ G) [AddSubgroup.Normal (AddMonoidHom.range f)] (x : G) : ↑x = 0 → x ∈ AddMonoidHom.range f

noncomputable def AddGroup.fintypeOfDomOfCoker {F : Type u} {G : Type u} [AddGroup F] [AddGroup G] [Fintype F] (f : F →+ G) [AddSubgroup.Normal (AddMonoidHom.range f)] [Fintype (G ⧸ AddMonoidHom.range f)] : Fintype G

If F and coker(F →+ G) are finite, then G is finite.

Equations

• AddGroup.fintypeOfDomOfCoker f = AddGroup.fintypeOfKerLeRange f (QuotientAddGroup.mk' (AddMonoidHom.range f)) ⋯

noncomputable def Group.fintypeOfDomOfCoker {F : Type u} {G : Type u} [Group F] [Group G] [Fintype F] (f : F →* G) [Subgroup.Normal (MonoidHom.range f)] [Fintype (G ⧸ MonoidHom.range f)] : Fintype G

If F and coker(F →* G) are finite, then G is finite.

Equations

• Group.fintypeOfDomOfCoker f = Group.fintypeOfKerLeRange f (QuotientGroup.mk' (MonoidHom.range f)) ⋯