File: Mathlib/GroupTheory/Subgroup/Center.lean

full_name: AddSubgroup.center
code: /-- The center of a group `G` is the set of elements that commute with everything in `G` -/
@[to_additive
      "The center of an additive group `G` is the set of elements that commute with
      everything in `G`"]
def center : Subgroup G 
goal: AddSubgroup G

full_name: Subgroup.center
code: /-- The center of a group `G` is the set of elements that commute with everything in `G` -/
@[to_additive
      "The center of an additive group `G` is the set of elements that commute with
      everything in `G`"]
def center : Subgroup G 
goal: Subgroup G

full_name: AddSubgroup.coe_center
code: @[to_additive]
theorem coe_center : ↑(center G) = Set.center G 
goal: ↑(AddSubgroup.center G) = Set.addCenter G

full_name: Subgroup.coe_center
code: @[to_additive]
theorem coe_center : ↑(center G) = Set.center G 
goal: ↑(Subgroup.center G) = Set.center G

full_name: AddSubgroup.center_toAddSubmonoid
code: @[to_additive (attr := simp)]
theorem center_toSubmonoid : (center G).toSubmonoid = Submonoid.center G 
goal: (AddSubgroup.center G).toAddSubmonoid = AddSubmonoid.center G

full_name: Subgroup.center_toSubmonoid
code: @[to_additive (attr := simp)]
theorem center_toSubmonoid : (center G).toSubmonoid = Submonoid.center G 
goal: (Subgroup.center G).toSubmonoid = Submonoid.center G

full_name: Subgroup.center.isCommutative
code: instance center.isCommutative : (center G).IsCommutative 
goal: (Subgroup.center G).IsCommutative

full_name: Subgroup.centerUnitsEquivUnitsCenter
code: /-- For a group with zero, the center of the units is the same as the units of the center. -/
@[simps! apply_val_coe symm_apply_coe_val]
def centerUnitsEquivUnitsCenter (G₀ : Type*) [GroupWithZero G₀] :
    Subgroup.center (G₀ˣ) ≃* (Submonoid.center G₀)ˣ 
goal: ↥(Subgroup.center G₀ˣ) ≃* (↥(Submonoid.center G₀))ˣ

full_name: Subgroup.val_centerUnitsEquivUnitsCenter_apply_coe
code: /-- For a group with zero, the center of the units is the same as the units of the center. -/
@[simps! apply_val_coe symm_apply_coe_val]
def centerUnitsEquivUnitsCenter (G₀ : Type*) [GroupWithZero G₀] :
    Subgroup.center (G₀ˣ) ≃* (Submonoid.center G₀)ˣ 
goal: ↑↑((Subgroup.centerUnitsEquivUnitsCenter G₀) a) = ↑↑a

full_name: Subgroup.val_centerUnitsEquivUnitsCenter_symm_apply_coe
code: /-- For a group with zero, the center of the units is the same as the units of the center. -/
@[simps! apply_val_coe symm_apply_coe_val]
def centerUnitsEquivUnitsCenter (G₀ : Type*) [GroupWithZero G₀] :
    Subgroup.center (G₀ˣ) ≃* (Submonoid.center G₀)ˣ 
goal: ↑↑((Subgroup.centerUnitsEquivUnitsCenter G₀).symm u) = ↑↑u

full_name: AddSubgroup.mem_center_iff
code: @[to_additive]
theorem mem_center_iff {z : G} : z ∈ center G ↔ ∀ g, g * z = z * g 
goal: z ∈ AddSubgroup.center G ↔ ∀ (g : G), g + z = z + g

full_name: Subgroup.mem_center_iff
code: @[to_additive]
theorem mem_center_iff {z : G} : z ∈ center G ↔ ∀ g, g * z = z * g 
goal: z ∈ Subgroup.center G ↔ ∀ (g : G), g * z = z * g

full_name: Subgroup.decidableMemCenter
code: instance decidableMemCenter (z : G) [Decidable (∀ g, g * z = z * g)] : Decidable (z ∈ center G) 
goal: Decidable (z ∈ Subgroup.center G)

full_name: Subgroup.centerCharacteristic
code: @[to_additive]
instance centerCharacteristic : (center G).Characteristic 
goal: (Subgroup.center G).Characteristic

full_name: AddSubgroup.centerCharacteristic
code: @[to_additive]
instance centerCharacteristic : (center G).Characteristic 
goal: (AddSubgroup.center G).Characteristic

full_name: CommGroup.center_eq_top
code: theorem _root_.CommGroup.center_eq_top {G : Type*} [CommGroup G] : center G = ⊤ 
goal: Subgroup.center G = ⊤

full_name: Group.commGroupOfCenterEqTop
code: /-- A group is commutative if the center is the whole group -/
def _root_.Group.commGroupOfCenterEqTop (h : center G = ⊤) : CommGroup G 
goal: CommGroup G

full_name: Subgroup.center_le_normalizer
code: @[to_additive]
theorem center_le_normalizer : center G ≤ H.normalizer 
goal: Subgroup.center G ≤ H.normalizer

full_name: AddSubgroup.center_le_normalizer
code: @[to_additive]
theorem center_le_normalizer : center G ≤ H.normalizer 
goal: AddSubgroup.center G ≤ H.normalizer

full_name: IsConj.eq_of_left_mem_center
code: theorem eq_of_left_mem_center {g h : M} (H : IsConj g h) (Hg : g ∈ Set.center M) : g = h 
goal: g = h

full_name: IsConj.eq_of_right_mem_center
code: theorem eq_of_right_mem_center {g h : M} (H : IsConj g h) (Hh : h ∈ Set.center M) : g = h 
goal: g = h

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