Actions by Subgroups

These are just copies of the definitions about Submonoid starting from Submonoid.mulAction.

Tags

subgroup, subgroups

instance AddSubgroup.instAddAction {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] {S : AddSubgroup G} : AddAction { x : G // x ∈ S } α

The additive action by an add_subgroup is the action by the underlying AddGroup.

Equations

• AddSubgroup.instAddAction = inferInstanceAs (AddAction { x : G // x ∈ S.toAddSubmonoid } α)

instance Subgroup.instMulAction {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] {S : Subgroup G} : MulAction { x : G // x ∈ S } α

The action by a subgroup is the action by the underlying group.

Equations

• Subgroup.instMulAction = inferInstanceAs (MulAction { x : G // x ∈ S.toSubmonoid } α)

theorem AddSubgroup.vadd_def {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] {S : AddSubgroup G} (g : { x : G // x ∈ S }) (m : α) : g +ᵥ m = ↑g +ᵥ m

theorem Subgroup.smul_def {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] {S : Subgroup G} (g : { x : G // x ∈ S }) (m : α) : g • m = ↑g • m

@[simp]

theorem AddSubgroup.mk_vadd {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] {S : AddSubgroup G} (g : G) (hg : g ∈ S) (a : α) : ⟨g, hg⟩ +ᵥ a = g +ᵥ a

@[simp]

theorem Subgroup.mk_smul {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] {S : Subgroup G} (g : G) (hg : g ∈ S) (a : α) : ⟨g, hg⟩ • a = g • a

instance AddSubgroup.vaddCommClass_left {G : Type u_1} {α : Type u_2} {β : Type u_3} [AddGroup G] [AddAction G β] [VAdd α β] [VAddCommClass G α β] (S : AddSubgroup G) : VAddCommClass { x : G // x ∈ S } α β

Equations

• ⋯ = ⋯

instance Subgroup.smulCommClass_left {G : Type u_1} {α : Type u_2} {β : Type u_3} [Group G] [MulAction G β] [SMul α β] [SMulCommClass G α β] (S : Subgroup G) : SMulCommClass { x : G // x ∈ S } α β

Equations

• ⋯ = ⋯

instance AddSubgroup.vaddCommClass_right {G : Type u_1} {α : Type u_2} {β : Type u_3} [AddGroup G] [VAdd α β] [AddAction G β] [VAddCommClass α G β] (S : AddSubgroup G) : VAddCommClass α { x : G // x ∈ S } β

Equations

• ⋯ = ⋯

instance Subgroup.smulCommClass_right {G : Type u_1} {α : Type u_2} {β : Type u_3} [Group G] [SMul α β] [MulAction G β] [SMulCommClass α G β] (S : Subgroup G) : SMulCommClass α { x : G // x ∈ S } β

Equations

• ⋯ = ⋯

instance Subgroup.instIsScalarTowerSubtypeMem {G : Type u_1} {α : Type u_2} {β : Type u_3} [Group G] [SMul α β] [MulAction G α] [MulAction G β] [IsScalarTower G α β] (S : Subgroup G) : IsScalarTower { x : G // x ∈ S } α β

Note that this provides IsScalarTower S G G which is needed by smul_mul_assoc.

Equations

• ⋯ = ⋯

instance Subgroup.instFaithfulSMulSubtypeMem {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] [FaithfulSMul G α] (S : Subgroup G) : FaithfulSMul { x : G // x ∈ S } α

Equations

• ⋯ = ⋯

instance Subgroup.instDistribMulActionSubtypeMem {G : Type u_1} {α : Type u_2} [Group G] [AddMonoid α] [DistribMulAction G α] (S : Subgroup G) : DistribMulAction { x : G // x ∈ S } α

The action by a subgroup is the action by the underlying group.

Equations

• S.instDistribMulActionSubtypeMem = inferInstanceAs (DistribMulAction { x : G // x ∈ S.toSubmonoid } α)

instance Subgroup.instMulDistribMulActionSubtypeMem {G : Type u_1} {α : Type u_2} [Group G] [Monoid α] [MulDistribMulAction G α] (S : Subgroup G) : MulDistribMulAction { x : G // x ∈ S } α

The action by a subgroup is the action by the underlying group.

Equations

• S.instMulDistribMulActionSubtypeMem = inferInstanceAs (MulDistribMulAction { x : G // x ∈ S.toSubmonoid } α)

instance Subgroup.center.smulCommClass_left {G : Type u_1} [Group G] : SMulCommClass { x : G // x ∈ Subgroup.center G } G G

The center of a group acts commutatively on that group.

Equations

• ⋯ = ⋯

instance Subgroup.center.smulCommClass_right {G : Type u_1} [Group G] : SMulCommClass G { x : G // x ∈ Subgroup.center G } G

The center of a group acts commutatively on that group.

Equations

• ⋯ = ⋯