Unbundled subgroups (deprecated)

This file is deprecated, and is no longer imported by anything in mathlib other than other deprecated files, and test files. You should not need to import it.

This file defines unbundled multiplicative and additive subgroups. Instead of using this file, please use Subgroup G and AddSubgroup A, defined in GroupTheory.Subgroup.Basic.

Main definitions

IsAddSubgroup (S : Set A) : the predicate that S is the underlying subset of an additive subgroup of A. The bundled variant AddSubgroup A should be used in preference to this.

IsSubgroup (S : Set G) : the predicate that S is the underlying subset of a subgroup of G. The bundled variant Subgroup G should be used in preference to this.

Tags

subgroup, subgroups, IsSubgroup

structure IsAddSubgroup {A : Type u_3} [AddGroup A] (s : Set A) extends IsAddSubmonoid : Prop

s is an additive subgroup: a set containing 0 and closed under addition and negation.

• zero_mem : 0 ∈ s
• add_mem : ∀ {a b : A}, a ∈ s → b ∈ s → a + b ∈ s
• neg_mem : ∀ {a : A}, a ∈ s → -a ∈ s

  The proposition that s is closed under negation.

structure IsSubgroup {G : Type u_1} [Group G] (s : Set G) extends IsSubmonoid : Prop

s is a subgroup: a set containing 1 and closed under multiplication and inverse.

• one_mem : 1 ∈ s
• mul_mem : ∀ {a b : G}, a ∈ s → b ∈ s → a * b ∈ s
• inv_mem : ∀ {a : G}, a ∈ s → a⁻¹ ∈ s

  The proposition that s is closed under inverse.

theorem IsAddSubgroup.sub_mem {G : Type u_1} [AddGroup G] {s : Set G} (hs : IsAddSubgroup s) {x : G} {y : G} (hx : x ∈ s) (hy : y ∈ s) : x - y ∈ s

theorem IsSubgroup.div_mem {G : Type u_1} [Group G] {s : Set G} (hs : IsSubgroup s) {x : G} {y : G} (hx : x ∈ s) (hy : y ∈ s) : x / y ∈ s

theorem Additive.isAddSubgroup {G : Type u_1} [Group G] {s : Set G} (hs : IsSubgroup s) : IsAddSubgroup s

theorem Additive.isAddSubgroup_iff {G : Type u_1} [Group G] {s : Set G} : IsAddSubgroup s ↔ IsSubgroup s

theorem Multiplicative.isSubgroup {A : Type u_3} [AddGroup A] {s : Set A} (hs : IsAddSubgroup s) : IsSubgroup s

theorem Multiplicative.isSubgroup_iff {A : Type u_3} [AddGroup A] {s : Set A} : IsSubgroup s ↔ IsAddSubgroup s

theorem IsAddSubgroup.of_add_neg {G : Type u_1} [AddGroup G] (s : Set G) (one_mem : 0 ∈ s) (div_mem : ∀ {a b : G}, a ∈ s → b ∈ s → a + -b ∈ s) : IsAddSubgroup s

theorem IsSubgroup.of_div {G : Type u_1} [Group G] (s : Set G) (one_mem : 1 ∈ s) (div_mem : ∀ {a b : G}, a ∈ s → b ∈ s → a * b⁻¹ ∈ s) : IsSubgroup s

theorem IsAddSubgroup.of_sub {A : Type u_3} [AddGroup A] (s : Set A) (zero_mem : 0 ∈ s) (sub_mem : ∀ {a b : A}, a ∈ s → b ∈ s → a - b ∈ s) : IsAddSubgroup s

theorem IsAddSubgroup.inter {G : Type u_1} [AddGroup G] {s₁ : Set G} {s₂ : Set G} (hs₁ : IsAddSubgroup s₁) (hs₂ : IsAddSubgroup s₂) : IsAddSubgroup (s₁ ∩ s₂)

theorem IsSubgroup.inter {G : Type u_1} [Group G] {s₁ : Set G} {s₂ : Set G} (hs₁ : IsSubgroup s₁) (hs₂ : IsSubgroup s₂) : IsSubgroup (s₁ ∩ s₂)

theorem IsAddSubgroup.iInter {G : Type u_1} [AddGroup G] {ι : Sort u_4} {s : ι → Set G} (hs : ∀ (y : ι), IsAddSubgroup (s y)) : IsAddSubgroup (Set.iInter s)

theorem IsSubgroup.iInter {G : Type u_1} [Group G] {ι : Sort u_4} {s : ι → Set G} (hs : ∀ (y : ι), IsSubgroup (s y)) : IsSubgroup (Set.iInter s)

theorem isAddSubgroup_iUnion_of_directed {G : Type u_1} [AddGroup G] {ι : Type u_4} [Nonempty ι] {s : ι → Set G} (hs : ∀ (i : ι), IsAddSubgroup (s i)) (directed : ∀ (i j : ι), ∃ (k : ι), s i ⊆ s k ∧ s j ⊆ s k) : IsAddSubgroup (⋃ (i : ι), s i)

abbrev isAddSubgroup_iUnion_of_directed.match_1 {G : Type u_2} {ι : Type u_1} {s : ι → Set G} : ∀ {a : G} (motive : (∃ (i : ι), a ∈ s i) → Prop) (x : ∃ (i : ι), a ∈ s i), (∀ (i : ι) (hi : a ∈ s i), motive ⋯) → motive x

Equations

• ⋯ = ⋯

theorem isSubgroup_iUnion_of_directed {G : Type u_1} [Group G] {ι : Type u_4} [Nonempty ι] {s : ι → Set G} (hs : ∀ (i : ι), IsSubgroup (s i)) (directed : ∀ (i j : ι), ∃ (k : ι), s i ⊆ s k ∧ s j ⊆ s k) : IsSubgroup (⋃ (i : ι), s i)

theorem IsAddSubgroup.neg_mem_iff {G : Type u_1} {a : G} [AddGroup G] {s : Set G} (hs : IsAddSubgroup s) : -a ∈ s ↔ a ∈ s

theorem IsSubgroup.inv_mem_iff {G : Type u_1} {a : G} [Group G] {s : Set G} (hs : IsSubgroup s) : a⁻¹ ∈ s ↔ a ∈ s

theorem IsAddSubgroup.add_mem_cancel_right {G : Type u_1} {a : G} {b : G} [AddGroup G] {s : Set G} (hs : IsAddSubgroup s) (h : a ∈ s) : b + a ∈ s ↔ b ∈ s

theorem IsSubgroup.mul_mem_cancel_right {G : Type u_1} {a : G} {b : G} [Group G] {s : Set G} (hs : IsSubgroup s) (h : a ∈ s) : b * a ∈ s ↔ b ∈ s

theorem IsAddSubgroup.add_mem_cancel_left {G : Type u_1} {a : G} {b : G} [AddGroup G] {s : Set G} (hs : IsAddSubgroup s) (h : a ∈ s) : a + b ∈ s ↔ b ∈ s

theorem IsSubgroup.mul_mem_cancel_left {G : Type u_1} {a : G} {b : G} [Group G] {s : Set G} (hs : IsSubgroup s) (h : a ∈ s) : a * b ∈ s ↔ b ∈ s

structure IsNormalAddSubgroup {A : Type u_3} [AddGroup A] (s : Set A) extends IsAddSubgroup : Prop

IsNormalAddSubgroup (s : Set A) expresses the fact that s is a normal additive subgroup of the additive group A. Important: the preferred way to say this in Lean is via bundled subgroups S : AddSubgroup A and hs : S.normal, and not via this structure.

• zero_mem : 0 ∈ s
• add_mem : ∀ {a b : A}, a ∈ s → b ∈ s → a + b ∈ s
• neg_mem : ∀ {a : A}, a ∈ s → -a ∈ s
• normal : ∀ n ∈ s, ∀ (g : A), g + n + -g ∈ s

  The proposition that s is closed under (additive) conjugation.

structure IsNormalSubgroup {G : Type u_1} [Group G] (s : Set G) extends IsSubgroup : Prop

IsNormalSubgroup (s : Set G) expresses the fact that s is a normal subgroup of the group G. Important: the preferred way to say this in Lean is via bundled subgroups S : Subgroup G and not via this structure.

• one_mem : 1 ∈ s
• mul_mem : ∀ {a b : G}, a ∈ s → b ∈ s → a * b ∈ s
• inv_mem : ∀ {a : G}, a ∈ s → a⁻¹ ∈ s
• normal : ∀ n ∈ s, ∀ (g : G), g * n * g⁻¹ ∈ s

  The proposition that s is closed under conjugation.

theorem isNormalAddSubgroup_of_addCommGroup {G : Type u_1} [AddCommGroup G] {s : Set G} (hs : IsAddSubgroup s) : IsNormalAddSubgroup s

theorem isNormalSubgroup_of_commGroup {G : Type u_1} [CommGroup G] {s : Set G} (hs : IsSubgroup s) : IsNormalSubgroup s

theorem Additive.isNormalAddSubgroup {G : Type u_1} [Group G] {s : Set G} (hs : IsNormalSubgroup s) : IsNormalAddSubgroup s

theorem Additive.isNormalAddSubgroup_iff {G : Type u_1} [Group G] {s : Set G} : IsNormalAddSubgroup s ↔ IsNormalSubgroup s

theorem Multiplicative.isNormalSubgroup {A : Type u_3} [AddGroup A] {s : Set A} (hs : IsNormalAddSubgroup s) : IsNormalSubgroup s

theorem Multiplicative.isNormalSubgroup_iff {A : Type u_3} [AddGroup A] {s : Set A} : IsNormalSubgroup s ↔ IsNormalAddSubgroup s

theorem IsAddSubgroup.mem_norm_comm {G : Type u_1} [AddGroup G] {s : Set G} (hs : IsNormalAddSubgroup s) {a : G} {b : G} (hab : a + b ∈ s) : b + a ∈ s

theorem IsSubgroup.mem_norm_comm {G : Type u_1} [Group G] {s : Set G} (hs : IsNormalSubgroup s) {a : G} {b : G} (hab : a * b ∈ s) : b * a ∈ s

theorem IsAddSubgroup.mem_norm_comm_iff {G : Type u_1} [AddGroup G] {s : Set G} (hs : IsNormalAddSubgroup s) {a : G} {b : G} : a + b ∈ s ↔ b + a ∈ s

theorem IsSubgroup.mem_norm_comm_iff {G : Type u_1} [Group G] {s : Set G} (hs : IsNormalSubgroup s) {a : G} {b : G} : a * b ∈ s ↔ b * a ∈ s

def IsAddSubgroup.trivial (G : Type u_4) [AddGroup G] : Set G

the trivial additive subgroup

Equations

• IsAddSubgroup.trivial G = {0}

def IsSubgroup.trivial (G : Type u_4) [Group G] : Set G

The trivial subgroup

Equations

• IsSubgroup.trivial G = {1}

@[simp]

theorem IsAddSubgroup.mem_trivial {G : Type u_1} [AddGroup G] {g : G} : g ∈ IsAddSubgroup.trivial G ↔ g = 0

@[simp]

theorem IsSubgroup.mem_trivial {G : Type u_1} [Group G] {g : G} : g ∈ IsSubgroup.trivial G ↔ g = 1

theorem IsAddSubgroup.trivial_normal {G : Type u_1} [AddGroup G] : IsNormalAddSubgroup (IsAddSubgroup.trivial G)

theorem IsSubgroup.trivial_normal {G : Type u_1} [Group G] : IsNormalSubgroup (IsSubgroup.trivial G)

theorem IsAddSubgroup.eq_trivial_iff {G : Type u_1} [AddGroup G] {s : Set G} (hs : IsAddSubgroup s) : s = IsAddSubgroup.trivial G ↔ ∀ x ∈ s, x = 0

theorem IsSubgroup.eq_trivial_iff {G : Type u_1} [Group G] {s : Set G} (hs : IsSubgroup s) : s = IsSubgroup.trivial G ↔ ∀ x ∈ s, x = 1

theorem IsAddSubgroup.univ_addSubgroup {G : Type u_1} [AddGroup G] : IsNormalAddSubgroup Set.univ

theorem IsSubgroup.univ_subgroup {G : Type u_1} [Group G] : IsNormalSubgroup Set.univ

def IsAddSubgroup.addCenter (G : Type u_4) [AddGroup G] : Set G

The underlying set of the center of an additive group.

Equations

• IsAddSubgroup.addCenter G = {z : G | ∀ (g : G), g + z = z + g}

def IsSubgroup.center (G : Type u_4) [Group G] : Set G

The underlying set of the center of a group.

Equations

• IsSubgroup.center G = {z : G | ∀ (g : G), g * z = z * g}

theorem IsAddSubgroup.mem_add_center {G : Type u_1} [AddGroup G] {a : G} : a ∈ IsAddSubgroup.addCenter G ↔ ∀ (g : G), g + a = a + g

theorem IsSubgroup.mem_center {G : Type u_1} [Group G] {a : G} : a ∈ IsSubgroup.center G ↔ ∀ (g : G), g * a = a * g

theorem IsAddSubgroup.add_center_normal {G : Type u_1} [AddGroup G] : IsNormalAddSubgroup (IsAddSubgroup.addCenter G)

theorem IsSubgroup.center_normal {G : Type u_1} [Group G] : IsNormalSubgroup (IsSubgroup.center G)

def IsAddSubgroup.addNormalizer {G : Type u_1} [AddGroup G] (s : Set G) : Set G

The underlying set of the normalizer of a subset S : Set A of an additive group A. That is, the elements a : A such that a + S - a = S.

Equations

• IsAddSubgroup.addNormalizer s = {g : G | ∀ (n : G), n ∈ s ↔ g + n + -g ∈ s}

def IsSubgroup.normalizer {G : Type u_1} [Group G] (s : Set G) : Set G

The underlying set of the normalizer of a subset S : Set G of a group G. That is, the elements g : G such that g * S * g⁻¹ = S.

Equations

• IsSubgroup.normalizer s = {g : G | ∀ (n : G), n ∈ s ↔ g * n * g⁻¹ ∈ s}

theorem IsAddSubgroup.normalizer_isAddSubgroup {G : Type u_1} [AddGroup G] (s : Set G) : IsAddSubgroup (IsAddSubgroup.addNormalizer s)

theorem IsSubgroup.normalizer_isSubgroup {G : Type u_1} [Group G] (s : Set G) : IsSubgroup (IsSubgroup.normalizer s)

theorem IsAddSubgroup.subset_add_normalizer {G : Type u_1} [AddGroup G] {s : Set G} (hs : IsAddSubgroup s) : s ⊆ IsAddSubgroup.addNormalizer s

theorem IsSubgroup.subset_normalizer {G : Type u_1} [Group G] {s : Set G} (hs : IsSubgroup s) : s ⊆ IsSubgroup.normalizer s

def IsAddGroupHom.ker {G : Type u_1} {H : Type u_2} [AddGroup H] (f : G → H) : Set G

ker f : Set A is the underlying subset of the kernel of a map A → B

Equations

• IsAddGroupHom.ker f = f ⁻¹' IsAddSubgroup.trivial H

def IsGroupHom.ker {G : Type u_1} {H : Type u_2} [Group H] (f : G → H) : Set G

ker f : Set G is the underlying subset of the kernel of a map G → H.

Equations

• IsGroupHom.ker f = f ⁻¹' IsSubgroup.trivial H

theorem IsAddGroupHom.mem_ker {G : Type u_1} {H : Type u_2} [AddGroup H] (f : G → H) {x : G} : x ∈ IsAddGroupHom.ker f ↔ f x = 0

theorem IsGroupHom.mem_ker {G : Type u_1} {H : Type u_2} [Group H] (f : G → H) {x : G} : x ∈ IsGroupHom.ker f ↔ f x = 1

theorem IsAddGroupHom.zero_ker_neg {G : Type u_1} {H : Type u_2} [AddGroup G] [AddGroup H] {f : G → H} (hf : IsAddGroupHom f) {a : G} {b : G} (h : f (a + -b) = 0) : f a = f b

theorem IsGroupHom.one_ker_inv {G : Type u_1} {H : Type u_2} [Group G] [Group H] {f : G → H} (hf : IsGroupHom f) {a : G} {b : G} (h : f (a * b⁻¹) = 1) : f a = f b

theorem IsAddGroupHom.zero_ker_neg' {G : Type u_1} {H : Type u_2} [AddGroup G] [AddGroup H] {f : G → H} (hf : IsAddGroupHom f) {a : G} {b : G} (h : f (-a + b) = 0) : f a = f b

theorem IsGroupHom.one_ker_inv' {G : Type u_1} {H : Type u_2} [Group G] [Group H] {f : G → H} (hf : IsGroupHom f) {a : G} {b : G} (h : f (a⁻¹ * b) = 1) : f a = f b

theorem IsAddGroupHom.neg_ker_zero {G : Type u_1} {H : Type u_2} [AddGroup G] [AddGroup H] {f : G → H} (hf : IsAddGroupHom f) {a : G} {b : G} (h : f a = f b) : f (a + -b) = 0

theorem IsGroupHom.inv_ker_one {G : Type u_1} {H : Type u_2} [Group G] [Group H] {f : G → H} (hf : IsGroupHom f) {a : G} {b : G} (h : f a = f b) : f (a * b⁻¹) = 1

theorem IsAddGroupHom.neg_ker_zero' {G : Type u_1} {H : Type u_2} [AddGroup G] [AddGroup H] {f : G → H} (hf : IsAddGroupHom f) {a : G} {b : G} (h : f a = f b) : f (-a + b) = 0

theorem IsGroupHom.inv_ker_one' {G : Type u_1} {H : Type u_2} [Group G] [Group H] {f : G → H} (hf : IsGroupHom f) {a : G} {b : G} (h : f a = f b) : f (a⁻¹ * b) = 1

theorem IsAddGroupHom.zero_iff_ker_neg {G : Type u_1} {H : Type u_2} [AddGroup G] [AddGroup H] {f : G → H} (hf : IsAddGroupHom f) (a : G) (b : G) : f a = f b ↔ f (a + -b) = 0

theorem IsGroupHom.one_iff_ker_inv {G : Type u_1} {H : Type u_2} [Group G] [Group H] {f : G → H} (hf : IsGroupHom f) (a : G) (b : G) : f a = f b ↔ f (a * b⁻¹) = 1

theorem IsAddGroupHom.zero_iff_ker_neg' {G : Type u_1} {H : Type u_2} [AddGroup G] [AddGroup H] {f : G → H} (hf : IsAddGroupHom f) (a : G) (b : G) : f a = f b ↔ f (-a + b) = 0

theorem IsGroupHom.one_iff_ker_inv' {G : Type u_1} {H : Type u_2} [Group G] [Group H] {f : G → H} (hf : IsGroupHom f) (a : G) (b : G) : f a = f b ↔ f (a⁻¹ * b) = 1

theorem IsAddGroupHom.neg_iff_ker {G : Type u_1} {H : Type u_2} [AddGroup G] [AddGroup H] {f : G → H} (hf : IsAddGroupHom f) (a : G) (b : G) : f a = f b ↔ a + -b ∈ IsAddGroupHom.ker f

theorem IsGroupHom.inv_iff_ker {G : Type u_1} {H : Type u_2} [Group G] [Group H] {f : G → H} (hf : IsGroupHom f) (a : G) (b : G) : f a = f b ↔ a * b⁻¹ ∈ IsGroupHom.ker f

theorem IsAddGroupHom.neg_iff_ker' {G : Type u_1} {H : Type u_2} [AddGroup G] [AddGroup H] {f : G → H} (hf : IsAddGroupHom f) (a : G) (b : G) : f a = f b ↔ -a + b ∈ IsAddGroupHom.ker f

theorem IsGroupHom.inv_iff_ker' {G : Type u_1} {H : Type u_2} [Group G] [Group H] {f : G → H} (hf : IsGroupHom f) (a : G) (b : G) : f a = f b ↔ a⁻¹ * b ∈ IsGroupHom.ker f

abbrev IsAddGroupHom.image_addSubgroup.match_1 {G : Type u_2} {H : Type u_1} {f : G → H} {s : Set G} {a₂ : H} (motive : a₂ ∈ f '' s → Prop) : ∀ (x : a₂ ∈ f '' s), (∀ (b₂ : G) (hb₂ : b₂ ∈ s) (eq₂ : f b₂ = a₂), motive ⋯) → motive x

Equations

• ⋯ = ⋯

theorem IsAddGroupHom.image_addSubgroup {G : Type u_1} {H : Type u_2} [AddGroup G] [AddGroup H] {f : G → H} (hf : IsAddGroupHom f) {s : Set G} (hs : IsAddSubgroup s) : IsAddSubgroup (f '' s)

theorem IsGroupHom.image_subgroup {G : Type u_1} {H : Type u_2} [Group G] [Group H] {f : G → H} (hf : IsGroupHom f) {s : Set G} (hs : IsSubgroup s) : IsSubgroup (f '' s)

theorem IsAddGroupHom.range_addSubgroup {G : Type u_1} {H : Type u_2} [AddGroup G] [AddGroup H] {f : G → H} (hf : IsAddGroupHom f) : IsAddSubgroup (Set.range f)

theorem IsGroupHom.range_subgroup {G : Type u_1} {H : Type u_2} [Group G] [Group H] {f : G → H} (hf : IsGroupHom f) : IsSubgroup (Set.range f)

theorem IsAddGroupHom.preimage {G : Type u_1} {H : Type u_2} [AddGroup G] [AddGroup H] {f : G → H} (hf : IsAddGroupHom f) {s : Set H} (hs : IsAddSubgroup s) : IsAddSubgroup (f ⁻¹' s)

theorem IsGroupHom.preimage {G : Type u_1} {H : Type u_2} [Group G] [Group H] {f : G → H} (hf : IsGroupHom f) {s : Set H} (hs : IsSubgroup s) : IsSubgroup (f ⁻¹' s)

theorem IsAddGroupHom.preimage_normal {G : Type u_1} {H : Type u_2} [AddGroup G] [AddGroup H] {f : G → H} (hf : IsAddGroupHom f) {s : Set H} (hs : IsNormalAddSubgroup s) : IsNormalAddSubgroup (f ⁻¹' s)

theorem IsGroupHom.preimage_normal {G : Type u_1} {H : Type u_2} [Group G] [Group H] {f : G → H} (hf : IsGroupHom f) {s : Set H} (hs : IsNormalSubgroup s) : IsNormalSubgroup (f ⁻¹' s)

theorem IsAddGroupHom.isNormalAddSubgroup_ker {G : Type u_1} {H : Type u_2} [AddGroup G] [AddGroup H] {f : G → H} (hf : IsAddGroupHom f) : IsNormalAddSubgroup (IsAddGroupHom.ker f)

theorem IsGroupHom.isNormalSubgroup_ker {G : Type u_1} {H : Type u_2} [Group G] [Group H] {f : G → H} (hf : IsGroupHom f) : IsNormalSubgroup (IsGroupHom.ker f)

theorem IsAddGroupHom.injective_of_trivial_ker {G : Type u_1} {H : Type u_2} [AddGroup G] [AddGroup H] {f : G → H} (hf : IsAddGroupHom f) (h : IsAddGroupHom.ker f = IsAddSubgroup.trivial G) : Function.Injective f

theorem IsGroupHom.injective_of_trivial_ker {G : Type u_1} {H : Type u_2} [Group G] [Group H] {f : G → H} (hf : IsGroupHom f) (h : IsGroupHom.ker f = IsSubgroup.trivial G) : Function.Injective f

theorem IsAddGroupHom.trivial_ker_of_injective {G : Type u_1} {H : Type u_2} [AddGroup G] [AddGroup H] {f : G → H} (hf : IsAddGroupHom f) (h : Function.Injective f) : IsAddGroupHom.ker f = IsAddSubgroup.trivial G

theorem IsGroupHom.trivial_ker_of_injective {G : Type u_1} {H : Type u_2} [Group G] [Group H] {f : G → H} (hf : IsGroupHom f) (h : Function.Injective f) : IsGroupHom.ker f = IsSubgroup.trivial G

theorem IsAddGroupHom.injective_iff_trivial_ker {G : Type u_1} {H : Type u_2} [AddGroup G] [AddGroup H] {f : G → H} (hf : IsAddGroupHom f) : Function.Injective f ↔ IsAddGroupHom.ker f = IsAddSubgroup.trivial G

theorem IsGroupHom.injective_iff_trivial_ker {G : Type u_1} {H : Type u_2} [Group G] [Group H] {f : G → H} (hf : IsGroupHom f) : Function.Injective f ↔ IsGroupHom.ker f = IsSubgroup.trivial G

theorem IsAddGroupHom.trivial_ker_iff_eq_zero {G : Type u_1} {H : Type u_2} [AddGroup G] [AddGroup H] {f : G → H} (hf : IsAddGroupHom f) : IsAddGroupHom.ker f = IsAddSubgroup.trivial G ↔ ∀ (x : G), f x = 0 → x = 0

theorem IsGroupHom.trivial_ker_iff_eq_one {G : Type u_1} {H : Type u_2} [Group G] [Group H] {f : G → H} (hf : IsGroupHom f) : IsGroupHom.ker f = IsSubgroup.trivial G ↔ ∀ (x : G), f x = 1 → x = 1

inductive AddGroup.InClosure {A : Type u_3} [AddGroup A] (s : Set A) : A → Prop

If A is an additive group and s : Set A, then InClosure s : Set A is the underlying subset of the subgroup generated by s.

• basic: ∀ {A : Type u_3} [inst : AddGroup A] {s : Set A} {a : A}, a ∈ s → AddGroup.InClosure s a
• zero: ∀ {A : Type u_3} [inst : AddGroup A] {s : Set A}, AddGroup.InClosure s 0
• neg: ∀ {A : Type u_3} [inst : AddGroup A] {s : Set A} {a : A}, AddGroup.InClosure s a → AddGroup.InClosure s (-a)
• add: ∀ {A : Type u_3} [inst : AddGroup A] {s : Set A} {a b : A}, AddGroup.InClosure s a → AddGroup.InClosure s b → AddGroup.InClosure s (a + b)

inductive Group.InClosure {G : Type u_1} [Group G] (s : Set G) : G → Prop

If G is a group and s : Set G, then InClosure s : Set G is the underlying subset of the subgroup generated by s.

• basic: ∀ {G : Type u_1} [inst : Group G] {s : Set G} {a : G}, a ∈ s → Group.InClosure s a
• one: ∀ {G : Type u_1} [inst : Group G] {s : Set G}, Group.InClosure s 1
• inv: ∀ {G : Type u_1} [inst : Group G] {s : Set G} {a : G}, Group.InClosure s a → Group.InClosure s a⁻¹
• mul: ∀ {G : Type u_1} [inst : Group G] {s : Set G} {a b : G}, Group.InClosure s a → Group.InClosure s b → Group.InClosure s (a * b)

def AddGroup.closure {G : Type u_1} [AddGroup G] (s : Set G) : Set G

AddGroup.closure s is the additive subgroup generated by s, i.e., the smallest additive subgroup containing s.

Equations

• AddGroup.closure s = {a : G | AddGroup.InClosure s a}

def Group.closure {G : Type u_1} [Group G] (s : Set G) : Set G

Group.closure s is the subgroup generated by s, i.e. the smallest subgroup containing s.

Equations

• Group.closure s = {a : G | Group.InClosure s a}

theorem AddGroup.mem_closure {G : Type u_1} [AddGroup G] {s : Set G} {a : G} : a ∈ s → a ∈ AddGroup.closure s

theorem Group.mem_closure {G : Type u_1} [Group G] {s : Set G} {a : G} : a ∈ s → a ∈ Group.closure s

theorem AddGroup.closure.isAddSubgroup {G : Type u_1} [AddGroup G] (s : Set G) : IsAddSubgroup (AddGroup.closure s)

theorem Group.closure.isSubgroup {G : Type u_1} [Group G] (s : Set G) : IsSubgroup (Group.closure s)

theorem AddGroup.subset_closure {G : Type u_1} [AddGroup G] {s : Set G} : s ⊆ AddGroup.closure s

theorem Group.subset_closure {G : Type u_1} [Group G] {s : Set G} : s ⊆ Group.closure s

theorem AddGroup.closure_subset {G : Type u_1} [AddGroup G] {s : Set G} {t : Set G} (ht : IsAddSubgroup t) (h : s ⊆ t) : AddGroup.closure s ⊆ t

theorem Group.closure_subset {G : Type u_1} [Group G] {s : Set G} {t : Set G} (ht : IsSubgroup t) (h : s ⊆ t) : Group.closure s ⊆ t

theorem AddGroup.closure_subset_iff {G : Type u_1} [AddGroup G] {s : Set G} {t : Set G} (ht : IsAddSubgroup t) : AddGroup.closure s ⊆ t ↔ s ⊆ t

theorem Group.closure_subset_iff {G : Type u_1} [Group G] {s : Set G} {t : Set G} (ht : IsSubgroup t) : Group.closure s ⊆ t ↔ s ⊆ t

theorem AddGroup.closure_mono {G : Type u_1} [AddGroup G] {s : Set G} {t : Set G} (h : s ⊆ t) : AddGroup.closure s ⊆ AddGroup.closure t

theorem Group.closure_mono {G : Type u_1} [Group G] {s : Set G} {t : Set G} (h : s ⊆ t) : Group.closure s ⊆ Group.closure t

@[simp]

theorem AddGroup.closure_addSubgroup {G : Type u_1} [AddGroup G] {s : Set G} (hs : IsAddSubgroup s) : AddGroup.closure s = s

@[simp]

theorem Group.closure_subgroup {G : Type u_1} [Group G] {s : Set G} (hs : IsSubgroup s) : Group.closure s = s

abbrev AddGroup.exists_list_of_mem_closure.match_1 {G : Type u_1} [AddGroup G] (L : List G) (x : G) (motive : (∃ a ∈ List.reverse L, -a = x) → Prop) : ∀ (x_1 : ∃ a ∈ List.reverse L, -a = x), (∀ (y : G) (hy1 : y ∈ List.reverse L) (hy2 : -y = x), motive ⋯) → motive x_1

Equations

• ⋯ = ⋯

abbrev AddGroup.exists_list_of_mem_closure.match_2 {G : Type u_1} [AddGroup G] {s : Set G} {x : G} (motive : (∃ (l : List G), (∀ x ∈ l, x ∈ s ∨ -x ∈ s) ∧ List.sum l = x) → Prop) : ∀ (x_1 : ∃ (l : List G), (∀ x ∈ l, x ∈ s ∨ -x ∈ s) ∧ List.sum l = x), (∀ (L : List G) (HL1 : ∀ x ∈ L, x ∈ s ∨ -x ∈ s) (HL2 : List.sum L = x), motive ⋯) → motive x_1

Equations

• ⋯ = ⋯

theorem AddGroup.exists_list_of_mem_closure {G : Type u_1} [AddGroup G] {s : Set G} {a : G} (h : a ∈ AddGroup.closure s) : ∃ (l : List G), (∀ x ∈ l, x ∈ s ∨ -x ∈ s) ∧ List.sum l = a

theorem Group.exists_list_of_mem_closure {G : Type u_1} [Group G] {s : Set G} {a : G} (h : a ∈ Group.closure s) : ∃ (l : List G), (∀ x ∈ l, x ∈ s ∨ x⁻¹ ∈ s) ∧ List.prod l = a

theorem AddGroup.image_closure {G : Type u_1} {H : Type u_2} [AddGroup G] [AddGroup H] {f : G → H} (hf : IsAddGroupHom f) (s : Set G) : f '' AddGroup.closure s = AddGroup.closure (f '' s)

theorem Group.image_closure {G : Type u_1} {H : Type u_2} [Group G] [Group H] {f : G → H} (hf : IsGroupHom f) (s : Set G) : f '' Group.closure s = Group.closure (f '' s)

theorem AddGroup.mclosure_subset {G : Type u_1} [AddGroup G] {s : Set G} : AddMonoid.Closure s ⊆ AddGroup.closure s

theorem Group.mclosure_subset {G : Type u_1} [Group G] {s : Set G} : Monoid.Closure s ⊆ Group.closure s

theorem AddGroup.mclosure_neg_subset {G : Type u_1} [AddGroup G] {s : Set G} : AddMonoid.Closure (Neg.neg ⁻¹' s) ⊆ AddGroup.closure s

theorem Group.mclosure_inv_subset {G : Type u_1} [Group G] {s : Set G} : Monoid.Closure (Inv.inv ⁻¹' s) ⊆ Group.closure s

theorem AddGroup.closure_eq_mclosure {G : Type u_1} [AddGroup G] {s : Set G} : AddGroup.closure s = AddMonoid.Closure (s ∪ Neg.neg ⁻¹' s)

theorem Group.closure_eq_mclosure {G : Type u_1} [Group G] {s : Set G} : Group.closure s = Monoid.Closure (s ∪ Inv.inv ⁻¹' s)

theorem AddGroup.mem_closure_union_iff {G : Type u_4} [AddCommGroup G] {s : Set G} {t : Set G} {x : G} : x ∈ AddGroup.closure (s ∪ t) ↔ ∃ y ∈ AddGroup.closure s, ∃ z ∈ AddGroup.closure t, y + z = x

theorem Group.mem_closure_union_iff {G : Type u_4} [CommGroup G] {s : Set G} {t : Set G} {x : G} : x ∈ Group.closure (s ∪ t) ↔ ∃ y ∈ Group.closure s, ∃ z ∈ Group.closure t, y * z = x

theorem IsAddSubgroup.trivial_eq_closure {G : Type u_1} [AddGroup G] : IsAddSubgroup.trivial G = AddGroup.closure ∅

theorem IsSubgroup.trivial_eq_closure {G : Type u_1} [Group G] : IsSubgroup.trivial G = Group.closure ∅

theorem Group.conjugatesOf_subset {G : Type u_1} [Group G] {t : Set G} (ht : IsNormalSubgroup t) {a : G} (h : a ∈ t) : conjugatesOf a ⊆ t

theorem Group.conjugatesOfSet_subset' {G : Type u_1} [Group G] {s : Set G} {t : Set G} (ht : IsNormalSubgroup t) (h : s ⊆ t) : Group.conjugatesOfSet s ⊆ t

def Group.normalClosure {G : Type u_1} [Group G] (s : Set G) : Set G

The normal closure of a set s is the subgroup closure of all the conjugates of elements of s. It is the smallest normal subgroup containing s.

Equations

• Group.normalClosure s = Group.closure (Group.conjugatesOfSet s)

theorem Group.conjugatesOfSet_subset_normalClosure {G : Type u_1} {s : Set G} [Group G] : Group.conjugatesOfSet s ⊆ Group.normalClosure s

theorem Group.subset_normalClosure {G : Type u_1} {s : Set G} [Group G] : s ⊆ Group.normalClosure s

theorem Group.normalClosure.isSubgroup {G : Type u_1} [Group G] (s : Set G) : IsSubgroup (Group.normalClosure s)

The normal closure of a set is a subgroup.

theorem Group.normalClosure.is_normal {G : Type u_1} {s : Set G} [Group G] : IsNormalSubgroup (Group.normalClosure s)

The normal closure of s is a normal subgroup.

theorem Group.normalClosure_subset {G : Type u_1} [Group G] {s : Set G} {t : Set G} (ht : IsNormalSubgroup t) (h : s ⊆ t) : Group.normalClosure s ⊆ t

The normal closure of s is the smallest normal subgroup containing s.

theorem Group.normalClosure_subset_iff {G : Type u_1} [Group G] {s : Set G} {t : Set G} (ht : IsNormalSubgroup t) : s ⊆ t ↔ Group.normalClosure s ⊆ t

theorem Group.normalClosure_mono {G : Type u_1} [Group G] {s : Set G} {t : Set G} : s ⊆ t → Group.normalClosure s ⊆ Group.normalClosure t

theorem AddSubgroup.of.proof_2 {G : Type u_1} [AddGroup G] {s : Set G} (h : IsAddSubgroup s) : 0 ∈ s

theorem AddSubgroup.of.proof_1 {G : Type u_1} [AddGroup G] {s : Set G} (h : IsAddSubgroup s) : ∀ {a b : G}, a ∈ s → b ∈ s → a + b ∈ s

def AddSubgroup.of {G : Type u_1} [AddGroup G] {s : Set G} (h : IsAddSubgroup s) : AddSubgroup G

Create a bundled additive subgroup from a set s and [IsAddSubgroup s].

Equations

• AddSubgroup.of h = { toAddSubmonoid := { toAddSubsemigroup := { carrier := s, add_mem' := ⋯ }, zero_mem' := ⋯ }, neg_mem' := ⋯ }

def Subgroup.of {G : Type u_1} [Group G] {s : Set G} (h : IsSubgroup s) : Subgroup G

Create a bundled subgroup from a set s and [IsSubgroup s].

Equations

• Subgroup.of h = { toSubmonoid := { toSubsemigroup := { carrier := s, mul_mem' := ⋯ }, one_mem' := ⋯ }, inv_mem' := ⋯ }

theorem AddSubgroup.isAddSubgroup {G : Type u_1} [AddGroup G] (K : AddSubgroup G) : IsAddSubgroup ↑K

theorem Subgroup.isSubgroup {G : Type u_1} [Group G] (K : Subgroup G) : IsSubgroup ↑K

theorem AddSubgroup.of_normal {G : Type u_1} [AddGroup G] (s : Set G) (h : IsAddSubgroup s) (n : IsNormalAddSubgroup s) : AddSubgroup.Normal (AddSubgroup.of h)

theorem Subgroup.of_normal {G : Type u_1} [Group G] (s : Set G) (h : IsSubgroup s) (n : IsNormalSubgroup s) : Subgroup.Normal (Subgroup.of h)