Centers of subgroups

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' := ⋯ }

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' := ⋯ }

theorem Subgroup.coe_center (G : Type u_1) [Group G] : ↑(Subgroup.center G) = Set.center G

theorem AddSubgroup.coe_center (G : Type u_1) [AddGroup G] : ↑(AddSubgroup.center G) = Set.addCenter G

@[simp]

theorem Subgroup.center_toSubmonoid (G : Type u_1) [Group G] : (Subgroup.center G).toSubmonoid = Submonoid.center G

@[simp]

theorem AddSubgroup.center_toAddSubmonoid (G : Type u_1) [AddGroup G] : (AddSubgroup.center G).toAddSubmonoid = AddSubmonoid.center G

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

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

The center of isomorphic groups are isomorphic.

Equations

• Subgroup.centerCongr e = Submonoid.centerCongr e

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

The center of isomorphic additive groups are isomorphic.

Equations

• AddSubgroup.centerCongr e = AddSubmonoid.centerCongr e

@[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)) : ↑((Subgroup.centerCongr e).symm s) = e.symm ↑s

@[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)) : ↑((AddSubgroup.centerCongr e).symm s) = e.symm ↑s

@[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)) : ↑((AddSubgroup.centerCongr e) r) = e ↑r

@[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)) : ↑((Subgroup.centerCongr e) r) = e ↑r

def Subgroup.centerToMulOpposite (G : Type u_1) [Group G] : ↥(Subgroup.center G) ≃* ↥(Subgroup.center Gᵐᵒᵖ)

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

Equations

• Subgroup.centerToMulOpposite G = Submonoid.centerToMulOpposite

def AddSubgroup.centerToAddOpposite (G : Type u_1) [AddGroup G] : ↥(AddSubgroup.center G) ≃+ ↥(AddSubgroup.center Gᵃᵒᵖ)

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

Equations

• AddSubgroup.centerToAddOpposite G = AddSubmonoid.centerToAddOpposite

@[simp]

theorem Subgroup.centerToMulOpposite_symm_apply_coe (G : Type u_1) [Group G] (r : ↥(Subsemigroup.center Gᵐᵒᵖ)) : ↑((Subgroup.centerToMulOpposite G).symm r) = MulOpposite.unop ↑r

@[simp]

theorem AddSubgroup.centerToAddOpposite_apply_coe (G : Type u_1) [AddGroup G] (r : ↥(AddSubsemigroup.center G)) : ↑((AddSubgroup.centerToAddOpposite G) r) = AddOpposite.op ↑r

@[simp]

theorem AddSubgroup.centerToAddOpposite_symm_apply_coe (G : Type u_1) [AddGroup G] (r : ↥(AddSubsemigroup.center Gᵃᵒᵖ)) : ↑((AddSubgroup.centerToAddOpposite G).symm r) = AddOpposite.unop ↑r

@[simp]

theorem Subgroup.centerToMulOpposite_apply_coe (G : Type u_1) [Group G] (r : ↥(Subsemigroup.center G)) : ↑((Subgroup.centerToMulOpposite G) r) = MulOpposite.op ↑r

def Subgroup.centerUnitsEquivUnitsCenter (G₀ : Type u_2) [GroupWithZero G₀] : ↥(Subgroup.center G₀ˣ) ≃* (↥(Submonoid.center G₀))ˣ

For a group with zero, the center of the units is the same as the units of the center.

Equations

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

@[simp]

theorem Subgroup.val_centerUnitsEquivUnitsCenter_apply_coe (G₀ : Type u_2) [GroupWithZero G₀] (a : ↥(Subgroup.center G₀ˣ)) : ↑↑((Subgroup.centerUnitsEquivUnitsCenter G₀) a) = ↑↑a

@[simp]

theorem Subgroup.val_centerUnitsEquivUnitsCenter_symm_apply_coe (G₀ : Type u_2) [GroupWithZero G₀] (u : (↥(Submonoid.center G₀))ˣ) : ↑↑((Subgroup.centerUnitsEquivUnitsCenter G₀).symm u) = ↑↑u

theorem Subgroup.mem_center_iff {G : Type u_1} [Group G] {z : G} : z ∈ Subgroup.center G ↔ ∀ (g : G), g * z = z * g

theorem AddSubgroup.mem_center_iff {G : Type u_1} [AddGroup G] {z : G} : z ∈ AddSubgroup.center G ↔ ∀ (g : G), g + z = z + g

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

Equations

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

instance Subgroup.centerCharacteristic {G : Type u_1} [Group G] : (Subgroup.center G).Characteristic

instance AddSubgroup.centerCharacteristic {G : Type u_1} [AddGroup G] : (AddSubgroup.center G).Characteristic

theorem CommGroup.center_eq_top {G : Type u_2} [CommGroup G] : Subgroup.center G = ⊤

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 ⋯

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

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

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

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

theorem ConjClasses.mk_bijOn (G : Type u_2) [Group G] : Set.BijOn ConjClasses.mk (↑(Subgroup.center G)) (ConjClasses.noncenter G)ᶜ