Simple groups

This file defines IsSimpleGroup G, a class indicating that a group has exactly two normal subgroups.

Main definitions

• IsSimpleGroup G, a class indicating that a group has exactly two normal subgroups.

Tags

subgroup, subgroups

class IsSimpleGroup (G : Type u_1) [Group G] extends Nontrivial G : Prop

A Group is simple when it has exactly two normal Subgroups.

• exists_pair_ne : ∃ (x : G) (y : G), x ≠ y
• eq_bot_or_eq_top_of_normal (H : Subgroup G) [H.Normal] : H = ⊥ ∨ H = ⊤

  Any normal subgroup is either ⊥ or ⊤

theorem isSimpleGroup_iff (G : Type u_1) [Group G] : IsSimpleGroup G ↔ Nontrivial G ∧ ∀ (H : Subgroup G) [H.Normal], H = ⊥ ∨ H = ⊤

class IsSimpleAddGroup (A : Type u_2) [AddGroup A] extends Nontrivial A : Prop

An AddGroup is simple when it has exactly two normal AddSubgroups.

• exists_pair_ne : ∃ (x : A) (y : A), x ≠ y
• eq_bot_or_eq_top_of_normal (H : AddSubgroup A) : H.Normal → H = ⊥ ∨ H = ⊤

  Any normal additive subgroup is either ⊥ or ⊤

theorem isSimpleAddGroup_iff (A : Type u_2) [AddGroup A] : IsSimpleAddGroup A ↔ Nontrivial A ∧ ∀ (H : AddSubgroup A), H.Normal → H = ⊥ ∨ H = ⊤

theorem Subgroup.Normal.eq_bot_or_eq_top {G : Type u_1} [Group G] [IsSimpleGroup G] (H : Subgroup G) [H.Normal] : H = ⊥ ∨ H = ⊤

theorem AddSubgroup.Normal.eq_bot_or_eq_top {G : Type u_1} [AddGroup G] [IsSimpleAddGroup G] (H : AddSubgroup G) [H.Normal] : H = ⊥ ∨ H = ⊤

theorem Subgroup.isSimpleGroup_iff {G : Type u_1} [Group G] {H : Subgroup G} : IsSimpleGroup ↥H ↔ H ≠ ⊥ ∧ ∀ H' ≤ H, (H'.subgroupOf H).Normal → H' = ⊥ ∨ H' = H

theorem AddSubgroup.isSimpleAddGroup_iff {G : Type u_1} [AddGroup G] {H : AddSubgroup G} : IsSimpleAddGroup ↥H ↔ H ≠ ⊥ ∧ ∀ H' ≤ H, (H'.addSubgroupOf H).Normal → H' = ⊥ ∨ H' = H

instance IsSimpleGroup.instIsSimpleOrderSubgroup {C : Type u_3} [CommGroup C] [IsSimpleGroup C] : IsSimpleOrder (Subgroup C)

instance IsSimpleAddGroup.instIsSimpleOrderAddSubgroup {C : Type u_3} [AddCommGroup C] [IsSimpleAddGroup C] : IsSimpleOrder (AddSubgroup C)

theorem IsSimpleGroup.of_isSimpleOrder {G : Type u_1} [Group G] [hG : IsSimpleOrder (Subgroup G)] : IsSimpleGroup G

A group whose subgroups are only ⊥ and ⊤ is simple.

theorem IsSimpleAddGroup.of_isSimpleOrder {G : Type u_1} [AddGroup G] [hG : IsSimpleOrder (AddSubgroup G)] : IsSimpleAddGroup G

An additive group whose subgroups are only ⊥ and ⊤ is simple.

theorem IsSimpleGroup.isSimpleGroup_of_surjective {G : Type u_1} [Group G] {H : Type u_3} [Group H] [IsSimpleGroup G] [Nontrivial H] (f : G →* H) (hf : Function.Surjective ⇑f) : IsSimpleGroup H

theorem IsSimpleAddGroup.isSimpleAddGroup_of_surjective {G : Type u_1} [AddGroup G] {H : Type u_3} [AddGroup H] [IsSimpleAddGroup G] [Nontrivial H] (f : G →+ H) (hf : Function.Surjective ⇑f) : IsSimpleAddGroup H

theorem MulEquiv.isSimpleGroup {G : Type u_1} [Group G] {H : Type u_3} [Group H] [IsSimpleGroup H] (e : G ≃* H) : IsSimpleGroup G

theorem AddEquiv.isSimpleAddGroup {G : Type u_1} [AddGroup G] {H : Type u_3} [AddGroup H] [IsSimpleAddGroup H] (e : G ≃+ H) : IsSimpleAddGroup G

theorem MulEquiv.isSimpleGroup_congr {G : Type u_1} [Group G] {H : Type u_3} [Group H] (e : G ≃* H) : IsSimpleGroup G ↔ IsSimpleGroup H

theorem AddEquiv.isSimpleAddGroup_congr {G : Type u_1} [AddGroup G] {H : Type u_3} [AddGroup H] (e : G ≃+ H) : IsSimpleAddGroup G ↔ IsSimpleAddGroup H