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

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

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

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

@[simp]

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

@[simp]

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

instance Subgroup.center.isMulCommutative (G : Type u_1) [Group G] : IsMulCommutative ↥(center G)

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.

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.

@[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

@[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

@[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

@[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

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.

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.

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

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

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

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

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

instance Subgroup.instNormalCenter {G : Type u_1} [Group G] : (center G).Normal

instance AddSubgroup.instNormalCenter {G : Type u_1} [AddGroup G] : (center G).Normal

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

theorem AddSubgroup.center_le_normalizer {G : Type u_1} [AddGroup G] {H : AddSubgroup G} : 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)) (noncenter G)ᶜ