Actions by Subgroups

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

Tags

subgroup, subgroups

@[instance_reducible]

instance Subgroup.instMulActionSubtypeMem {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] {S : Subgroup G} : MulAction (↥S) α

This shortcut instance provides a speedup.

Equations

• Subgroup.instMulActionSubtypeMem = inferInstance

@[instance_reducible]

instance AddSubgroup.instAddActionSubtypeMem {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] {S : AddSubgroup G} : AddAction (↥S) α

This shortcut instance provides a speedup.

Equations

• AddSubgroup.instAddActionSubtypeMem = inferInstance

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

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

@[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

@[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

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 (↥S) α β

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 (↥S) α β

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 (↥S) β

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 (↥S) β

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 (↥S) α β

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

instance AddSubgroup.instVAddAssocClassSubtypeMem {G : Type u_1} {α : Type u_2} {β : Type u_3} [AddGroup G] [VAdd α β] [AddAction G α] [AddAction G β] [VAddAssocClass G α β] (S : AddSubgroup G) : VAddAssocClass (↥S) α β

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

instance AddSubgroup.instFaithfulVAddSubtypeMem {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] [FaithfulVAdd G α] (S : AddSubgroup G) : FaithfulVAdd (↥S) α

@[instance_reducible]

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

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

Equations

• S.instDistribMulActionSubtypeMem = { toMulAction := Subgroup.instMulActionSubtypeMem, smul_zero := ⋯, smul_add := ⋯ }

@[instance_reducible]

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

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

Equations

• S.instMulDistribMulActionSubtypeMem = { toMulAction := Subgroup.instMulActionSubtypeMem, smul_one := ⋯, smul_mul := ⋯ }

instance Subgroup.center.smulCommClass_left {G : Type u_1} [Group G] : SMulCommClass (↥(center G)) G G

The center of a group acts commutatively on that group.

instance Subgroup.center.smulCommClass_right {G : Type u_1} [Group G] : SMulCommClass G (↥(center G)) G

The center of a group acts commutatively on that group.

theorem MonoidWithZeroHom.comap_mker {M : Type u_1} {N : Type u_2} {P : Type u_3} [MulZeroOneClass M] [MulZeroOneClass N] [MulZeroOneClass P] (g : N →*₀ P) (f : M →*₀ N) : Submonoid.comap f (MonoidHom.mker g) = MonoidHom.mker (g.comp f)