The orbit-stabilizer theorem for concrete groups

Content created by Jonathan Prieto-Cubides, Fredrik Bakke and Egbert Rijke.

Created on 2022-07-10.
Last modified on 2023-05-16.

module group-theory.orbit-stabilizer-theorem-concrete-groups where
Imports 
open import foundation.dependent-pair-types
open import foundation.universe-levels

open import group-theory.concrete-group-actions
open import group-theory.concrete-groups
open import group-theory.mere-equivalences-concrete-group-actions
open import group-theory.stabilizer-groups-concrete-group-actions

open import structured-types.pointed-types

Idea

The orbit stabilizer theorem for concrete groups asserts that the type Orbit(x) of orbits of an element x : X * is deloopable and fits in a fiber sequence

  BG_x ----> BG ----> B(Orbit(x))

To see that this is indeed a formulation of the orbit-stabilizer theorem, note that the delooping of Orbit(x) gives Orbit(x) the structure of a group. Furthermore, this fiber sequence induces a short exact sequence

  G_x ----> G ----> Orbit(x),

which induces a bijection from the cosets of the stabilizer subgroup G_x of G to the type Orbit(x).

Definitions

module _
  {l1 l2 : Level} (G : Concrete-Group l1) (X : action-Concrete-Group l2 G)
  where

  classifying-type-quotient-stabilizer-action-Concrete-Group :
    type-action-Concrete-Group G X → UU (lsuc l1 ⊔ lsuc l2)
  classifying-type-quotient-stabilizer-action-Concrete-Group x =
    Σ ( action-Concrete-Group (l1 ⊔ l2) G)
      ( mere-equiv-action-Concrete-Group G
        ( action-stabilizer-action-Concrete-Group G X x))

  point-classifying-type-quotient-stabilizer-action-Concrete-Group :
    (x : type-action-Concrete-Group G X) →
    classifying-type-quotient-stabilizer-action-Concrete-Group x
  pr1 (point-classifying-type-quotient-stabilizer-action-Concrete-Group x) =
    action-stabilizer-action-Concrete-Group G X x
  pr2 (point-classifying-type-quotient-stabilizer-action-Concrete-Group x) =
    refl-mere-equiv-action-Concrete-Group G
      (action-stabilizer-action-Concrete-Group G X x)

  classifying-pointed-type-stabilizer-action-Concrete-Group :
    (x : type-action-Concrete-Group G X) →
    Pointed-Type (lsuc l1 ⊔ lsuc l2)
  pr1 (classifying-pointed-type-stabilizer-action-Concrete-Group x) =
    classifying-type-quotient-stabilizer-action-Concrete-Group x
  pr2 (classifying-pointed-type-stabilizer-action-Concrete-Group x) =
    point-classifying-type-quotient-stabilizer-action-Concrete-Group x

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