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

The congruence relation generated by a normal subgroup.

Equations

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

def QuotientAddGroup.con {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : N.Normal] : 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 : N.Normal] {a : G} {b : G} {c : G} {d : G} (hab : (QuotientAddGroup.leftRel N) a b) (hcd : (QuotientAddGroup.leftRel N) c d) : (QuotientAddGroup.leftRel N) (a + c) (b + d)

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

Equations

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

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

Equations

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

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

The group homomorphism from G to G/N.

Equations

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

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

The additive group homomorphism from G to G/N.

Equations

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

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

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

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

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

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

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

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

theorem QuotientAddGroup.addMonoidHom_ext {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : N.Normal] {M : Type x} [AddMonoid M] ⦃f : G ⧸ N →+ M⦄ ⦃g : G ⧸ N →+ M⦄ (h : f.comp (QuotientAddGroup.mk' N) = g.comp (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 : N.Normal] {M : Type x} [Monoid M] ⦃f : G ⧸ N →* M⦄ ⦃g : G ⧸ N →* M⦄ (h : f.comp (QuotientGroup.mk' N) = g.comp (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 QuotientGroup.eq_one_iff {G : Type u} [Group G] {N : Subgroup G} [N.Normal] (x : G) : ↑x = 1 ↔ x ∈ N

@[simp]

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

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

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

@[simp]

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

@[simp]

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

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

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

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

Equations

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

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

theorem QuotientGroup.strictMono_comap_prod_map {G : Type u} [Group G] (N : Subgroup G) [nN : N.Normal] : StrictMono fun (H : Subgroup G) => (Subgroup.comap N.subtype H, Subgroup.map (QuotientGroup.mk' N) H)

theorem QuotientAddGroup.strictMono_comap_prod_map {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : N.Normal] : StrictMono fun (H : AddSubgroup G) => (AddSubgroup.comap N.subtype H, AddSubgroup.map (QuotientAddGroup.mk' N) H)

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

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

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

def QuotientGroup.map {G : Type u} [Group G] (N : Subgroup G) [nN : N.Normal] {H : Type v} [Group H] (M : Subgroup H) [M.Normal] (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 ((QuotientGroup.mk' M).comp 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 QuotientAddGroup.map {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : N.Normal] {H : Type v} [AddGroup H] (M : AddSubgroup H) [M.Normal] (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 ((QuotientAddGroup.mk' M).comp f) ⋯

@[simp]

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

@[simp]

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

theorem QuotientGroup.map_mk' {G : Type u} [Group G] (N : Subgroup G) [nN : N.Normal] {H : Type v} [Group H] (M : Subgroup H) [M.Normal] (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_mk' {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : N.Normal] {H : Type v} [AddGroup H] (M : AddSubgroup H) [M.Normal] (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_id_apply {G : Type u} [Group G] (N : Subgroup G) [nN : N.Normal] (h : optParam (N ≤ Subgroup.comap (MonoidHom.id G) N) ⋯) (x : G ⧸ N) : (QuotientGroup.map N N (MonoidHom.id G) h) x = x

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

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

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

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

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

def QuotientGroup.congr {G : Type u} [Group G] {H : Type v} [Group H] (G' : Subgroup G) (H' : Subgroup H) [G'.Normal] [H'.Normal] (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

• QuotientGroup.congr G' H' e he = { toFun := ⇑(QuotientGroup.map G' H' ↑e ⋯), invFun := ⇑(QuotientGroup.map H' G' ↑e.symm ⋯), left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯ }

theorem QuotientAddGroup.congr.proof_8 {G : Type u_1} [AddGroup G] {H : Type u_2} [AddGroup H] (G' : AddSubgroup G) (H' : AddSubgroup H) [G'.Normal] [H'.Normal] (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_4 {G : Type u_1} [AddGroup G] {H : Type u_2} [AddGroup H] : AddMonoidHomClass (H ≃+ G) H G

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

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

def QuotientAddGroup.congr {G : Type u} [AddGroup G] {H : Type v} [AddGroup H] (G' : AddSubgroup G) (H' : AddSubgroup H) [G'.Normal] [H'.Normal] (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

• QuotientAddGroup.congr G' H' e he = { toFun := ⇑(QuotientAddGroup.map G' H' ↑e ⋯), invFun := ⇑(QuotientAddGroup.map H' G' ↑e.symm ⋯), left_inv := ⋯, right_inv := ⋯, map_add' := ⋯ }

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

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'

@[simp]

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

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

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

Equations

• QuotientGroup.kerLift φ = QuotientGroup.lift φ.ker φ ⋯

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

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

Equations

• QuotientAddGroup.kerLift φ = QuotientAddGroup.lift φ.ker φ ⋯

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

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

@[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

@[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

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

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

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

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

Equations

• QuotientGroup.rangeKerLift φ = QuotientGroup.lift φ.ker φ.rangeRestrict ⋯

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

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

Equations

• QuotientAddGroup.rangeKerLift φ = QuotientAddGroup.lift φ.ker φ.rangeRestrict ⋯

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

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

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

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

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.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)

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

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

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_1 {G : Type u_1} [AddGroup G] {H : Type u_2} [AddGroup H] (φ : G →+ H) : φ.ker.Normal

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 ⧸ φ.ker ≃* H

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

Equations

• QuotientGroup.quotientKerEquivOfRightInverse φ ψ hφ = { toFun := ⇑(QuotientGroup.kerLift φ), invFun := QuotientGroup.mk ∘ ψ, left_inv := ⋯, right_inv := hφ, map_mul' := ⋯ }

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

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 ⧸ φ.ker ≃+ H

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

Equations

• QuotientAddGroup.quotientKerEquivOfRightInverse φ ψ hφ = { toFun := ⇑(QuotientAddGroup.kerLift φ), invFun := QuotientAddGroup.mk ∘ ψ, left_inv := ⋯, right_inv := hφ, map_add' := ⋯ }

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

@[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), (QuotientGroup.quotientKerEquivOfRightInverse φ ψ hφ).symm 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 ⧸ φ.ker) : (QuotientGroup.quotientKerEquivOfRightInverse φ ψ hφ) a = (QuotientGroup.kerLift φ) a

@[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 ⧸ φ.ker) : (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), (QuotientAddGroup.quotientKerEquivOfRightInverse φ ψ hφ).symm a = (QuotientAddGroup.mk ∘ ψ) 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.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 QuotientAddGroup.quotientBot_symm_apply {G : Type u} [AddGroup G] : ∀ (a : G), QuotientAddGroup.quotientBot.symm a = ↑a

@[simp]

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

@[simp]

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

@[simp]

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

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

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

def QuotientGroup.quotientMulEquivOfEq {G : Type u} [Group G] {M : Subgroup G} {N : Subgroup G} [M.Normal] [N.Normal] (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 = { toEquiv := Subgroup.quotientEquivOfEq h, map_mul' := ⋯ }

def QuotientAddGroup.quotientAddEquivOfEq {G : Type u} [AddGroup G] {M : AddSubgroup G} {N : AddSubgroup G} [M.Normal] [N.Normal] (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 = { toEquiv := AddSubgroup.quotientEquivOfEq h, map_add' := ⋯ }

theorem QuotientAddGroup.quotientAddEquivOfEq.proof_1 {G : Type u_1} [AddGroup G] {M : AddSubgroup G} {N : AddSubgroup G} [M.Normal] [N.Normal] (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

@[simp]

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

@[simp]

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

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

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

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 A.subtype A' ≤ AddSubgroup.comap A.subtype B'

@[simp]

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

@[simp]

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

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

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

Equations

• QuotientAddGroup.equivQuotientAddSubgroupOfOfEq h' h = (QuotientAddGroup.quotientMapAddSubgroupOfOfLe ⋯ ⋯).toAddEquiv (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 : (A'.addSubgroupOf A).Normal] [hBN : (B'.addSubgroupOf B).Normal] (h' : A' = B') (h : A = B) : (QuotientAddGroup.quotientMapAddSubgroupOfOfLe ⋯ ⋯).comp (QuotientAddGroup.quotientMapAddSubgroupOfOfLe ⋯ ⋯) = AddMonoidHom.id (↥B ⧸ B'.addSubgroupOf 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_4 {G : Type u_1} [AddGroup G] {A : AddSubgroup G} {B : AddSubgroup G} (h : A = B) : B ≤ A

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

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

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

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

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

Equations

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

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 ∈ (zsmulAddGroupHom n).range → g ∈ ((QuotientAddGroup.mk' (zsmulAddGroupHom n).range).comp f).ker

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

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

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

Equations

• QuotientAddGroup.homQuotientZSMulOfHom f n = QuotientAddGroup.lift (zsmulAddGroupHom n).range ((QuotientAddGroup.mk' (zsmulAddGroupHom n).range).comp f) ⋯

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

@[simp]

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

@[simp]

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

@[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 (f.comp g) n = (QuotientGroup.homQuotientZPowOfHom f n).comp (QuotientGroup.homQuotientZPowOfHom g 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 (f.comp g) n = (QuotientAddGroup.homQuotientZSMulOfHom f n).comp (QuotientAddGroup.homQuotientZSMulOfHom g 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) : (QuotientGroup.homQuotientZPowOfHom f n).comp (QuotientGroup.homQuotientZPowOfHom g n) = MonoidHom.id (B ⧸ (zpowGroupHom n).range)

@[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) : (QuotientAddGroup.homQuotientZSMulOfHom f n).comp (QuotientAddGroup.homQuotientZSMulOfHom g n) = AddMonoidHom.id (B ⧸ (zsmulAddGroupHom n).range)

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

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

Equations

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

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

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) : (QuotientAddGroup.homQuotientZSMulOfHom (↑e) n).comp (QuotientAddGroup.homQuotientZSMulOfHom (↑e.symm) n) = AddMonoidHom.id (B ⧸ (zsmulAddGroupHom n).range)

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

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

Equations

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

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

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

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

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

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

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

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

noncomputable def QuotientAddGroup.quotientInfEquivSumNormalQuotient {G : Type u} [AddGroup G] (H : AddSubgroup G) (N : AddSubgroup G) [N.Normal] : ↥H ⧸ N.addSubgroupOf H ≃+ ↥(H ⊔ N) ⧸ N.addSubgroupOf (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_4 {G : Type u_1} [AddGroup G] (H : AddSubgroup G) (N : AddSubgroup G) [N.Normal] : ((QuotientAddGroup.mk' (N.addSubgroupOf (H ⊔ N))).comp (AddSubgroup.inclusion ⋯)).ker.Normal

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

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

Equations

• ⋯ = ⋯

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

Equations

• ⋯ = ⋯

def QuotientGroup.quotientQuotientEquivQuotientAux {G : Type u} [Group G] (N : Subgroup G) [nN : N.Normal] (M : Subgroup G) [nM : M.Normal] (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) ⋯

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

@[simp]

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

@[simp]

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

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

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

def QuotientGroup.quotientQuotientEquivQuotient {G : Type u} [Group G] (N : Subgroup G) [nN : N.Normal] (M : Subgroup G) [nM : M.Normal] (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.

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

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

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

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

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

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.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.comap_comap_center {G : Type u} [Group G] {H₁ : Subgroup G} [H₁.Normal] {H₂ : Subgroup (G ⧸ H₁)} [H₂.Normal] : 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 QuotientAddGroup.comap_comap_center {G : Type u} [AddGroup G] {H₁ : AddSubgroup G} [H₁.Normal] {H₂ : AddSubgroup (G ⧸ H₁)} [H₂.Normal] : 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))

@[simp]

theorem QuotientAddGroup.mk_nat_mul {R : Type u_1} [NonAssocRing R] (N : AddSubgroup R) [N.Normal] (n : ℕ) (a : R) : ↑(↑n * a) = n • ↑a

@[simp]

theorem QuotientAddGroup.mk_int_mul {R : Type u_1} [NonAssocRing R] (N : AddSubgroup R) [N.Normal] (n : ℤ) (a : R) : ↑(↑n * a) = n • ↑a