# Documentation

Mathlib.GroupTheory.Subgroup.Actions

# Actions by Subgroups #

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

## Tags #

subgroup, subgroups

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

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

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theorem AddSubgroup.vadd_def {G : Type u_1} [inst : ] {α : Type u_2} [inst : ] {S : } (g : { x // x S }) (m : α) :
g +ᵥ m = g +ᵥ m
theorem Subgroup.smul_def {G : Type u_1} [inst : ] {α : Type u_2} [inst : ] {S : } (g : { x // x S }) (m : α) :
g m = g m
def AddSubgroup.vaddCommClass_left.proof_1 {G : Type u_1} [inst : ] {α : Type u_2} {β : Type u_3} [inst : ] [inst : VAdd α β] [inst : ] (S : ) :
VAddCommClass { x // x S.toAddSubmonoid } α β
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instance AddSubgroup.vaddCommClass_left {G : Type u_1} [inst : ] {α : Type u_2} {β : Type u_3} [inst : ] [inst : VAdd α β] [inst : ] (S : ) :
VAddCommClass { x // x S } α β
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instance Subgroup.smulCommClass_left {G : Type u_1} [inst : ] {α : Type u_2} {β : Type u_3} [inst : ] [inst : SMul α β] [inst : ] (S : ) :
SMulCommClass { x // x S } α β
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instance AddSubgroup.vaddCommClass_right {G : Type u_1} [inst : ] {α : Type u_2} {β : Type u_3} [inst : VAdd α β] [inst : ] [inst : ] (S : ) :
VAddCommClass α { x // x S } β
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def AddSubgroup.vaddCommClass_right.proof_1 {G : Type u_1} [inst : ] {α : Type u_2} {β : Type u_3} [inst : VAdd α β] [inst : ] [inst : ] (S : ) :
VAddCommClass α { x // x S.toAddSubmonoid } β
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instance Subgroup.smulCommClass_right {G : Type u_1} [inst : ] {α : Type u_2} {β : Type u_3} [inst : SMul α β] [inst : ] [inst : ] (S : ) :
SMulCommClass α { x // x S } β
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instance Subgroup.instIsScalarTowerSubtypeMemSubgroupInstMembershipInstSetLikeSubgroupSmulToMulOneClassToMonoidToDivInvMonoidToSMulToSubmonoidSmulToSMul {G : Type u_1} [inst : ] {α : Type u_2} {β : Type u_3} [inst : SMul α β] [inst : ] [inst : ] [inst : ] (S : ) :
IsScalarTower { x // x S } α β

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

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

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

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• One or more equations did not get rendered due to their size.
instance Subgroup.center.smulCommClass_left {G : Type u_1} [inst : ] :
SMulCommClass { x // } G G

The center of a group acts commutatively on that group.

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

The center of a group acts commutatively on that group.

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