Documentation

Mathlib.GroupTheory.GroupAction.Basic

Basic properties of group actions #

This file primarily concerns itself with orbits, stabilizers, and other objects defined in terms of actions. Despite this file being called basic, low-level helper lemmas for algebraic manipulation of • belong elsewhere.

Main definitions #

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theorem MulAction.fst_mem_orbit_of_mem_orbit {M : Type u} [Monoid M] {α : Type v} [MulAction M α] {β : Type u_1} [MulAction M β] {x y : α × β} (h : x ∈ orbit M y) :
x.1 ∈ orbit M y.1
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theorem AddAction.fst_mem_orbit_of_mem_orbit {M : Type u} [AddMonoid M] {α : Type v} [AddAction M α] {β : Type u_1} [AddAction M β] {x y : α × β} (h : x ∈ orbit M y) :
x.1 ∈ orbit M y.1
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theorem MulAction.snd_mem_orbit_of_mem_orbit {M : Type u} [Monoid M] {α : Type v} [MulAction M α] {β : Type u_1} [MulAction M β] {x y : α × β} (h : x ∈ orbit M y) :
x.2 ∈ orbit M y.2
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theorem AddAction.snd_mem_orbit_of_mem_orbit {M : Type u} [AddMonoid M] {α : Type v} [AddAction M α] {β : Type u_1} [AddAction M β] {x y : α × β} (h : x ∈ orbit M y) :
x.2 ∈ orbit M y.2
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theorem Finite.finite_mulAction_orbit {M : Type u} [Monoid M] {α : Type v} [MulAction M α] [Finite M] (a : α) :
(MulAction.orbit M a).Finite
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theorem Finite.finite_addAction_orbit {M : Type u} [AddMonoid M] {α : Type v} [AddAction M α] [Finite M] (a : α) :
(AddAction.orbit M a).Finite
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theorem MulAction.orbit_eq_univ (M : Type u) [Monoid M] {α : Type v} [MulAction M α] [IsPretransitive M α] (a : α) :
orbit M a = Set.univ
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theorem AddAction.orbit_eq_univ (M : Type u) [AddMonoid M] {α : Type v} [AddAction M α] [IsPretransitive M α] (a : α) :
orbit M a = Set.univ
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@[simp]
theorem MulAction.subsingleton_orbit_iff_mem_fixedPoints {M : Type u} [Monoid M] {α : Type v} [MulAction M α] {a : α} :
(orbit M a).Subsingleton ↔ a ∈ fixedPoints M α
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@[simp]
theorem AddAction.subsingleton_orbit_iff_mem_fixedPoints {M : Type u} [AddMonoid M] {α : Type v} [AddAction M α] {a : α} :
(orbit M a).Subsingleton ↔ a ∈ fixedPoints M α
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theorem MulAction.mem_fixedPoints_iff_card_orbit_eq_one {M : Type u} [Monoid M] {α : Type v} [MulAction M α] {a : α} [Fintype ↑(orbit M a)] :
a ∈ fixedPoints M α ↔ Fintype.card ↑(orbit M a) = 1
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theorem AddAction.mem_fixedPoints_iff_card_orbit_eq_one {M : Type u} [AddMonoid M] {α : Type v} [AddAction M α] {a : α} [Fintype ↑(orbit M a)] :
a ∈ fixedPoints M α ↔ Fintype.card ↑(orbit M a) = 1
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instance MulAction.instDecidablePredMemSetFixedByOfDecidableEq {M : Type u} [Monoid M] {β : Type u_1} [MulAction M β] (m : M) [DecidableEq β] :
DecidablePred fun (b : β) => b ∈ fixedBy β m
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instance AddAction.instDecidablePredMemSetFixedByAddOfDecidableEq {M : Type u} [AddMonoid M] {β : Type u_1} [AddAction M β] (m : M) [DecidableEq β] :
DecidablePred fun (b : β) => b ∈ fixedBy β m
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theorem smul_cancel_of_non_zero_divisor {M : Type u_1} {R : Type u_2} [Monoid M] [NonUnitalNonAssocRing R] [DistribMulAction M R] (k : M) (h : ∀ (x : R), k • x = 0 → x = 0) {a b : R} (h' : k • a = k • b) :
a = b

smul by a k : M over a ring is injective, if k is not a zero divisor. The general theory of such k is elaborated by IsSMulRegular. The typeclass that restricts all terms of M to have this property is NoZeroSMulDivisors.

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theorem MulAction.fixedPoints_of_subsingleton {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] [Subsingleton α] :
fixedPoints G α = Set.univ
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theorem AddAction.fixedPoints_of_subsingleton {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] [Subsingleton α] :
fixedPoints G α = Set.univ
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theorem MulAction.nontrivial_of_fixedPoints_ne_univ {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] (h : fixedPoints G α ≠ Set.univ) :
Nontrivial α

If a group acts nontrivially, then the type is nontrivial

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theorem AddAction.nontrivial_of_fixedPoints_ne_univ {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] (h : fixedPoints G α ≠ Set.univ) :
Nontrivial α

If a subgroup acts nontrivially, then the type is nontrivial.

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@[simp]
theorem MulAction.smul_orbit {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] (g : G) (a : α) :
g • orbit G a = orbit G a
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@[simp]
theorem AddAction.vadd_orbit {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] (g : G) (a : α) :
g +ᵥ orbit G a = orbit G a
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instance MulAction.instIsPretransitiveElemOrbit {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] (a : α) :
IsPretransitive G ↑(orbit G a)

The action of a group on an orbit is transitive.

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instance AddAction.instIsPretransitiveElemOrbit {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] (a : α) :
IsPretransitive G ↑(orbit G a)

The action of an additive group on an orbit is transitive.

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theorem MulAction.orbitRel_subgroup_le {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] (H : Subgroup G) :
orbitRel (↥H) α ≤ orbitRel G α
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theorem AddAction.orbitRel_addSubgroup_le {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] (H : AddSubgroup G) :
orbitRel (↥H) α ≤ orbitRel G α
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theorem MulAction.orbitRel_subgroupOf {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] (H K : Subgroup G) :
orbitRel (↥(H.subgroupOf K)) α = orbitRel (↥(H ⊓ K)) α
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theorem AddAction.orbitRel_addSubgroupOf {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] (H K : AddSubgroup G) :
orbitRel (↥(H.addSubgroupOf K)) α = orbitRel (↥(H ⊓ K)) α
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theorem MulAction.pretransitive_iff_subsingleton_quotient (G : Type u_1) (α : Type u_2) [Group G] [MulAction G α] :
IsPretransitive G α ↔ Subsingleton (orbitRel.Quotient G α)

An action is pretransitive if and only if the quotient by MulAction.orbitRel is a subsingleton.

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theorem AddAction.pretransitive_iff_subsingleton_quotient (G : Type u_1) (α : Type u_2) [AddGroup G] [AddAction G α] :
IsPretransitive G α ↔ Subsingleton (orbitRel.Quotient G α)

An additive action is pretransitive if and only if the quotient by AddAction.orbitRel is a subsingleton.

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theorem MulAction.pretransitive_iff_unique_quotient_of_nonempty (G : Type u_1) (α : Type u_2) [Group G] [MulAction G α] [Nonempty α] :
IsPretransitive G α ↔ Nonempty (Unique (orbitRel.Quotient G α))

If α is non-empty, an action is pretransitive if and only if the quotient has exactly one element.

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theorem AddAction.pretransitive_iff_unique_quotient_of_nonempty (G : Type u_1) (α : Type u_2) [AddGroup G] [AddAction G α] [Nonempty α] :
IsPretransitive G α ↔ Nonempty (Unique (orbitRel.Quotient G α))

If α is non-empty, an additive action is pretransitive if and only if the quotient has exactly one element.

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instance MulAction.instIsPretransitiveElemOrbit_1 {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] (x : orbitRel.Quotient G α) :
IsPretransitive G ↑x.orbit
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instance AddAction.instIsPretransitiveElemOrbit_1 {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] (x : orbitRel.Quotient G α) :
IsPretransitive G ↑x.orbit
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theorem Finite.of_finite_mulAction_orbitRel_quotient (G : Type u_1) (α : Type u_2) [Group G] [MulAction G α] [Finite G] [Finite (MulAction.orbitRel.Quotient G α)] :
Finite α
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theorem Finite.of_finite_addAction_orbitRel_quotient (G : Type u_1) (α : Type u_2) [AddGroup G] [AddAction G α] [Finite G] [Finite (AddAction.orbitRel.Quotient G α)] :
Finite α
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theorem MulAction.orbitRel_le_fst (G : Type u_1) (α : Type u_2) (β : Type u_3) [Group G] [MulAction G α] [MulAction G β] :
orbitRel G (α × β) ≤ Setoid.comap Prod.fst (orbitRel G α)
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theorem AddAction.orbitRel_le_fst (G : Type u_1) (α : Type u_2) (β : Type u_3) [AddGroup G] [AddAction G α] [AddAction G β] :
orbitRel G (α × β) ≤ Setoid.comap Prod.fst (orbitRel G α)
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theorem MulAction.orbitRel_le_snd (G : Type u_1) (α : Type u_2) (β : Type u_3) [Group G] [MulAction G α] [MulAction G β] :
orbitRel G (α × β) ≤ Setoid.comap Prod.snd (orbitRel G β)
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theorem AddAction.orbitRel_le_snd (G : Type u_1) (α : Type u_2) (β : Type u_3) [AddGroup G] [AddAction G α] [AddAction G β] :
orbitRel G (α × β) ≤ Setoid.comap Prod.snd (orbitRel G β)
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theorem MulAction.stabilizer_smul_eq_stabilizer_map_conj {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] (g : G) (a : α) :
stabilizer G (g • a) = Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) (stabilizer G a)

If the stabilizer of a is S, then the stabilizer of g • a is gSg⁻¹.

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noncomputable def MulAction.stabilizerEquivStabilizerOfOrbitRel {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] {a b : α} (h : (orbitRel G α) a b) :
↥(stabilizer G a) ≃* ↥(stabilizer G b)

A bijection between the stabilizers of two elements in the same orbit.

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    theorem AddAction.stabilizer_vadd_eq_stabilizer_map_conj {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] (g : G) (a : α) :
    stabilizer G (g +ᵥ a) = AddSubgroup.map (AddEquiv.toAddMonoidHom (AddAut.conj g)) (stabilizer G a)

    If the stabilizer of x is S, then the stabilizer of g +ᵥ x is g + S + (-g).

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    noncomputable def AddAction.stabilizerEquivStabilizerOfOrbitRel {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] {a b : α} (h : (orbitRel G α) a b) :
    ↥(stabilizer G a) ≃+ ↥(stabilizer G b)

    A bijection between the stabilizers of two elements in the same orbit.

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      theorem Equiv.swap_mem_stabilizer {α : Type u_1} [DecidableEq α] {S : Set α} {a b : α} :
      swap a b ∈ MulAction.stabilizer (Perm α) S ↔ (a ∈ S ↔ b ∈ S)
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      theorem MulAction.le_stabilizer_iff_smul_le {G : Type u_1} [Group G] {α : Type u_2} [MulAction G α] (s : Set α) (H : Subgroup G) :
      H ≤ stabilizer G s ↔ ∀ g ∈ H, g • s ⊆ s

      To prove inclusion of a subgroup in a stabilizer, it is enough to prove inclusions.

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      theorem AddAction.le_stabilizer_iff_vadd_le {G : Type u_1} [AddGroup G] {α : Type u_2} [AddAction G α] (s : Set α) (H : AddSubgroup G) :
      H ≤ stabilizer G s ↔ ∀ g ∈ H, g +ᵥ s ⊆ s

      To prove inclusion of a subgroup in a stabilizer, it is enough to prove inclusions.

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      theorem Module.stabilizer_units_eq_bot_of_ne_zero (R : Type u_1) {M : Type u_2} [Ring R] [AddCommGroup M] [Module R M] [NoZeroSMulDivisors R M] {x : M} (hx : x ≠ 0) :
      MulAction.stabilizer Rˣ x = ⊥