File: Mathlib/GroupTheory/Subgroup/Centralizer.lean

full_name: Subgroup.centralizer
code: /-- The `centralizer` of `H` is the subgroup of `g : G` commuting with every `h : H`. -/
@[to_additive
      "The `centralizer` of `H` is the additive subgroup of `g : G` commuting with every `h : H`."]
def centralizer (s : Set G) : Subgroup G 
goal: Subgroup G

full_name: AddSubgroup.centralizer
code: /-- The `centralizer` of `H` is the subgroup of `g : G` commuting with every `h : H`. -/
@[to_additive
      "The `centralizer` of `H` is the additive subgroup of `g : G` commuting with every `h : H`."]
def centralizer (s : Set G) : Subgroup G 
goal: AddSubgroup G

full_name: Subgroup.mem_centralizer_iff
code: @[to_additive]
theorem mem_centralizer_iff {g : G} {s : Set G} : g ∈ centralizer s ↔ ∀ h ∈ s, h * g = g * h 
goal: g ∈ Subgroup.centralizer s ↔ ∀ h ∈ s, h * g = g * h

full_name: AddSubgroup.mem_centralizer_iff
code: @[to_additive]
theorem mem_centralizer_iff {g : G} {s : Set G} : g ∈ centralizer s ↔ ∀ h ∈ s, h * g = g * h 
goal: g ∈ AddSubgroup.centralizer s ↔ ∀ h ∈ s, h + g = g + h

full_name: AddSubgroup.mem_centralizer_iff_commutator_eq_zero
code: @[to_additive]
theorem mem_centralizer_iff_commutator_eq_one {g : G} {s : Set G} :
    g ∈ centralizer s ↔ ∀ h ∈ s, h * g * h⁻¹ * g⁻¹ = 1 
goal: g ∈ AddSubgroup.centralizer s ↔ ∀ h ∈ s, h + g + -h + -g = 0

full_name: Subgroup.mem_centralizer_iff_commutator_eq_one
code: @[to_additive]
theorem mem_centralizer_iff_commutator_eq_one {g : G} {s : Set G} :
    g ∈ centralizer s ↔ ∀ h ∈ s, h * g * h⁻¹ * g⁻¹ = 1 
goal: g ∈ Subgroup.centralizer s ↔ ∀ h ∈ s, h * g * h⁻¹ * g⁻¹ = 1

full_name: AddSubgroup.centralizer_univ
code: @[to_additive]
theorem centralizer_univ : centralizer Set.univ = center G 
goal: AddSubgroup.centralizer Set.univ = AddSubgroup.center G

full_name: Subgroup.centralizer_univ
code: @[to_additive]
theorem centralizer_univ : centralizer Set.univ = center G 
goal: Subgroup.centralizer Set.univ = Subgroup.center G

full_name: AddSubgroup.le_centralizer_iff
code: @[to_additive]
theorem le_centralizer_iff : H ≤ centralizer K ↔ K ≤ centralizer H 
goal: H ≤ AddSubgroup.centralizer ↑K ↔ K ≤ AddSubgroup.centralizer ↑H

full_name: Subgroup.le_centralizer_iff
code: @[to_additive]
theorem le_centralizer_iff : H ≤ centralizer K ↔ K ≤ centralizer H 
goal: H ≤ Subgroup.centralizer ↑K ↔ K ≤ Subgroup.centralizer ↑H

full_name: AddSubgroup.center_le_centralizer
code: @[to_additive]
theorem center_le_centralizer (s) : center G ≤ centralizer s 
goal: AddSubgroup.center G ≤ AddSubgroup.centralizer s

full_name: Subgroup.center_le_centralizer
code: @[to_additive]
theorem center_le_centralizer (s) : center G ≤ centralizer s 
goal: Subgroup.center G ≤ Subgroup.centralizer s

full_name: Subgroup.centralizer_le
code: @[to_additive]
theorem centralizer_le {s t : Set G} (h : s ⊆ t) : centralizer t ≤ centralizer s 
goal: Subgroup.centralizer t ≤ Subgroup.centralizer s

full_name: AddSubgroup.centralizer_le
code: @[to_additive]
theorem centralizer_le {s t : Set G} (h : s ⊆ t) : centralizer t ≤ centralizer s 
goal: AddSubgroup.centralizer t ≤ AddSubgroup.centralizer s

full_name: Subgroup.centralizer_eq_top_iff_subset
code: @[to_additive (attr := simp)]
theorem centralizer_eq_top_iff_subset {s : Set G} : centralizer s = ⊤ ↔ s ⊆ center G 
goal: Subgroup.centralizer s = ⊤ ↔ s ⊆ ↑(Subgroup.center G)

full_name: AddSubgroup.centralizer_eq_top_iff_subset
code: @[to_additive (attr := simp)]
theorem centralizer_eq_top_iff_subset {s : Set G} : centralizer s = ⊤ ↔ s ⊆ center G 
goal: AddSubgroup.centralizer s = ⊤ ↔ s ⊆ ↑(AddSubgroup.center G)

full_name: AddSubgroup.Centralizer.characteristic
code: @[to_additive]
instance Centralizer.characteristic [hH : H.Characteristic] :
    (centralizer (H : Set G)).Characteristic 
goal: (AddSubgroup.centralizer ↑H).Characteristic

full_name: Subgroup.Centralizer.characteristic
code: @[to_additive]
instance Centralizer.characteristic [hH : H.Characteristic] :
    (centralizer (H : Set G)).Characteristic 
goal: (Subgroup.centralizer ↑H).Characteristic

full_name: Subgroup.le_centralizer_iff_isCommutative
code: @[to_additive]
theorem le_centralizer_iff_isCommutative : K ≤ centralizer K ↔ K.IsCommutative 
goal: K ≤ Subgroup.centralizer ↑K ↔ K.IsCommutative

full_name: AddSubgroup.le_centralizer_iff_isCommutative
code: @[to_additive]
theorem le_centralizer_iff_isCommutative : K ≤ centralizer K ↔ K.IsCommutative 
goal: K ≤ AddSubgroup.centralizer ↑K ↔ K.IsCommutative

full_name: Subgroup.le_centralizer
code: @[to_additive]
theorem le_centralizer [h : H.IsCommutative] : H ≤ centralizer H 
goal: H ≤ Subgroup.centralizer ↑H

full_name: AddSubgroup.le_centralizer
code: @[to_additive]
theorem le_centralizer [h : H.IsCommutative] : H ≤ centralizer H 
goal: H ≤ AddSubgroup.centralizer ↑H