Documentation

Mathlib.GroupTheory.Subgroup.Simple

Simple groups #

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

Main definitions #

Tags #

subgroup, subgroups

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

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.NormalH = H =

    Any normal subgroup is either or

Instances
    theorem IsSimpleGroup.eq_bot_or_eq_top_of_normal {G : Type u_1} :
    ∀ {inst : Group G} [self : IsSimpleGroup G] (H : Subgroup G), H.NormalH = H =

    Any normal subgroup is either or

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

    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.NormalH = H =

      Any normal additive subgroup is either or

    Instances
      theorem IsSimpleAddGroup.eq_bot_or_eq_top_of_normal {A : Type u_2} :
      ∀ {inst : AddGroup A} [self : IsSimpleAddGroup A] (H : AddSubgroup A), H.NormalH = H =

      Any normal additive subgroup is either or

      theorem AddSubgroup.Normal.eq_bot_or_eq_top {G : Type u_1} [AddGroup G] [IsSimpleAddGroup G] {H : AddSubgroup G} (Hn : H.Normal) :
      H = H =
      theorem Subgroup.Normal.eq_bot_or_eq_top {G : Type u_1} [Group G] [IsSimpleGroup G] {H : Subgroup G} (Hn : H.Normal) :
      H = H =