Centers of subgroups #

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def Subgroup.center (G : Type u_1) [Group G] :

Subgroup G

The center of a group G is the set of elements that commute with everything in G

Equations

Subgroup.center G = { carrier := Set.center G, mul_mem' := ⋯, one_mem' := ⋯, inv_mem' := ⋯ }

Instances For

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def AddSubgroup.center (G : Type u_1) [AddGroup G] :

AddSubgroup G

The center of an additive group G is the set of elements that commute with everything in G

Equations

AddSubgroup.center G = { carrier := Set.addCenter G, add_mem' := ⋯, zero_mem' := ⋯, neg_mem' := ⋯ }

Instances For

AddSubgroup.centerCharacteristic

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theorem Subgroup.coe_center (G : Type u_1) [Group G] :

↑(center G) = Set.center G

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theorem AddSubgroup.coe_center (G : Type u_1) [AddGroup G] :

↑(center G) = Set.addCenter G

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@[simp]

theorem Subgroup.center_toSubmonoid (G : Type u_1) [Group G] :

(center G).toSubmonoid = Submonoid.center G

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@[simp]

theorem AddSubgroup.center_toAddSubmonoid (G : Type u_1) [AddGroup G] :

(center G).toAddSubmonoid = AddSubmonoid.center G

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instance Subgroup.center.isCommutative (G : Type u_1) [Group G] :

(center G).IsCommutative

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def Subgroup.centerCongr {G : Type u_1} [Group G] {H : Type u_2} [Group H] (e : G ≃* H) :

↥(center G) ≃* ↥(center H)

The center of isomorphic groups are isomorphic.

Equations

Subgroup.centerCongr e = Submonoid.centerCongr e

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def AddSubgroup.centerCongr {G : Type u_1} [AddGroup G] {H : Type u_2} [AddGroup H] (e : G ≃+ H) :

↥(center G) ≃+ ↥(center H)

The center of isomorphic additive groups are isomorphic.

Equations

AddSubgroup.centerCongr e = AddSubmonoid.centerCongr e

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@[simp]

theorem Subgroup.centerCongr_symm_apply_coe {G : Type u_1} [Group G] {H : Type u_2} [Group H] (e : G ≃* H) (s : ↥(Subsemigroup.center H)) :

↑((centerCongr e).symm s) = e.symm ↑s

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@[simp]

theorem AddSubgroup.centerCongr_apply_coe {G : Type u_1} [AddGroup G] {H : Type u_2} [AddGroup H] (e : G ≃+ H) (r : ↥(AddSubsemigroup.center G)) :

↑((centerCongr e) r) = e ↑r

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@[simp]

theorem Subgroup.centerCongr_apply_coe {G : Type u_1} [Group G] {H : Type u_2} [Group H] (e : G ≃* H) (r : ↥(Subsemigroup.center G)) :

↑((centerCongr e) r) = e ↑r

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@[simp]

theorem AddSubgroup.centerCongr_symm_apply_coe {G : Type u_1} [AddGroup G] {H : Type u_2} [AddGroup H] (e : G ≃+ H) (s : ↥(AddSubsemigroup.center H)) :

↑((centerCongr e).symm s) = e.symm ↑s

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def Subgroup.centerToMulOpposite (G : Type u_1) [Group G] :

↥(center G) ≃* ↥(center Gᵐᵒᵖ)

The center of a group is isomorphic to the center of its opposite.

Equations

Subgroup.centerToMulOpposite G = Submonoid.centerToMulOpposite

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def AddSubgroup.centerToAddOpposite (G : Type u_1) [AddGroup G] :

↥(center G) ≃+ ↥(center Gᵃᵒᵖ)

The center of an additive group is isomorphic to the center of its opposite.

Equations

AddSubgroup.centerToAddOpposite G = AddSubmonoid.centerToAddOpposite

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@[simp]

theorem Subgroup.centerToMulOpposite_symm_apply_coe (G : Type u_1) [Group G] (r : ↥(Subsemigroup.center Gᵐᵒᵖ)) :

↑((centerToMulOpposite G).symm r) = MulOpposite.unop ↑r

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@[simp]

theorem Subgroup.centerToMulOpposite_apply_coe (G : Type u_1) [Group G] (r : ↥(Subsemigroup.center G)) :

↑((centerToMulOpposite G) r) = MulOpposite.op ↑r

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@[simp]

theorem AddSubgroup.centerToAddOpposite_apply_coe (G : Type u_1) [AddGroup G] (r : ↥(AddSubsemigroup.center G)) :

↑((centerToAddOpposite G) r) = AddOpposite.op ↑r

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@[simp]

theorem AddSubgroup.centerToAddOpposite_symm_apply_coe (G : Type u_1) [AddGroup G] (r : ↥(AddSubsemigroup.center Gᵃᵒᵖ)) :

↑((centerToAddOpposite G).symm r) = AddOpposite.unop ↑r

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theorem Subgroup.mem_center_iff {G : Type u_1} [Group G] {z : G} :

z ∈ center G ↔ ∀ (g : G), g * z = z * g

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theorem AddSubgroup.mem_center_iff {G : Type u_1} [AddGroup G] {z : G} :

z ∈ center G ↔ ∀ (g : G), g + z = z + g

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instance Subgroup.decidableMemCenter {G : Type u_1} [Group G] (z : G) [Decidable (∀ (g : G), g * z = z * g)] :

Decidable (z ∈ center G)

Equations

Subgroup.decidableMemCenter z = decidable_of_iff' (∀ (g : G), g * z = z * g) ⋯

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instance Subgroup.centerCharacteristic {G : Type u_1} [Group G] :

(center G).Characteristic

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instance AddSubgroup.centerCharacteristic {G : Type u_1} [AddGroup G] :

(center G).Characteristic

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theorem CommGroup.center_eq_top {G : Type u_2} [CommGroup G] :

Subgroup.center G = ⊤

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def Group.commGroupOfCenterEqTop {G : Type u_1} [Group G] (h : Subgroup.center G = ⊤) :

CommGroup G

A group is commutative if the center is the whole group

Equations

Group.commGroupOfCenterEqTop h = CommGroup.mk ⋯

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theorem Subgroup.center_le_normalizer {G : Type u_1} [Group G] {H : Subgroup G} :

center G ≤ H.normalizer

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theorem AddSubgroup.center_le_normalizer {G : Type u_1} [AddGroup G] {H : AddSubgroup G} :

center G ≤ H.normalizer

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theorem IsConj.eq_of_left_mem_center {M : Type u_2} [Monoid M] {g h : M} (H : IsConj g h) (Hg : g ∈ Set.center M) :

g = h

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theorem IsConj.eq_of_right_mem_center {M : Type u_2} [Monoid M] {g h : M} (H : IsConj g h) (Hh : h ∈ Set.center M) :

g = h

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theorem ConjClasses.mk_bijOn (G : Type u_2) [Group G] :

Set.BijOn ConjClasses.mk (↑(Subgroup.center G)) (noncenter G)ᶜ

Documentation

Mathlib.GroupTheory.Subgroup.Center

Centers of subgroups #