Documentation

Mathlib.GroupTheory.Subgroup.Center

Centers of subgroups #

theorem AddSubgroup.center.proof_1 (G : Type u_1) [AddGroup G] :
∀ {a b : G}, a (AddSubmonoid.center G).carrierb (AddSubmonoid.center G).carriera + b (AddSubmonoid.center G).carrier

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

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    theorem AddSubgroup.center.proof_3 (G : Type u_1) [AddGroup G] :
    ∀ {x : G}, x Set.addCenter G-x Set.addCenter G
    def Subgroup.center (G : Type u_1) [Group G] :

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

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      @[simp]
      theorem Subgroup.center_toSubmonoid (G : Type u_1) [Group G] :
      instance Subgroup.center.isCommutative (G : Type u_1) [Group G] :
      (Subgroup.center G).IsCommutative
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      @[simp]
      theorem Subgroup.val_centerUnitsEquivUnitsCenter_apply_coe (G₀ : Type u_2) [GroupWithZero G₀] (a : { x : G₀ˣ // x Subgroup.center G₀ˣ }) :
      ((Subgroup.centerUnitsEquivUnitsCenter G₀) a) = a
      @[simp]
      theorem Subgroup.val_centerUnitsEquivUnitsCenter_symm_apply_coe (G₀ : Type u_2) [GroupWithZero G₀] (u : { x : G₀ // x Submonoid.center G₀ }ˣ) :
      ((Subgroup.centerUnitsEquivUnitsCenter G₀).symm u) = u
      def Subgroup.centerUnitsEquivUnitsCenter (G₀ : Type u_2) [GroupWithZero G₀] :
      { x : G₀ˣ // x Subgroup.center G₀ˣ } ≃* { x : G₀ // x Submonoid.center G₀ }ˣ

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

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      • One or more equations did not get rendered due to their size.
      Instances For
        theorem AddSubgroup.mem_center_iff {G : Type u_1} [AddGroup G] {z : G} :
        z AddSubgroup.center G ∀ (g : G), g + z = z + g
        theorem Subgroup.mem_center_iff {G : Type u_1} [Group G] {z : G} :
        z Subgroup.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)] :
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        instance AddSubgroup.centerCharacteristic {G : Type u_1} [AddGroup G] :
        (AddSubgroup.center G).Characteristic
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        • =
        instance Subgroup.centerCharacteristic {G : Type u_1} [Group G] :
        (Subgroup.center G).Characteristic
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        A group is commutative if the center is the whole group

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          theorem Subgroup.center_le_normalizer {G : Type u_1} [Group G] {H : Subgroup G} :
          Subgroup.center G H.normalizer
          theorem IsConj.eq_of_left_mem_center {M : Type u_2} [Monoid M] {g : M} {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 : M} {h : M} (H : IsConj g h) (Hh : h Set.center M) :
          g = h
          theorem ConjClasses.mk_bijOn (G : Type u_2) [Group G] :