Group actions

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

Created on 2022-03-17.
Last modified on 2023-11-24.

module where
open import foundation.action-on-identifications-functions
open import foundation.dependent-pair-types
open import foundation.equivalences
open import foundation.function-extensionality
open import foundation.function-types
open import foundation.homotopies
open import foundation.identity-types
open import foundation.sets
open import foundation.universe-levels

open import group-theory.groups
open import group-theory.homomorphisms-groups
open import group-theory.subgroups
open import group-theory.symmetric-groups
open import group-theory.trivial-group-homomorphisms


An action of a group G on a set X, also called a G-action, is a group homomorphism from G into symmetric-Group X. A set equipped with a G-action is called a G-set.


The type of G-sets

module _
  {l1 : Level} (G : Group l1)

  action-Group : (l : Level)  UU (l1  lsuc l)
  action-Group l =
    Σ (Set l)  X  hom-Group G (symmetric-Group X))

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

  set-action-Group : Set l2
  set-action-Group = pr1 X

  type-action-Group : UU l2
  type-action-Group = type-Set set-action-Group

  is-set-type-action-Group : is-set type-action-Group
  is-set-type-action-Group = is-set-type-Set set-action-Group

  equiv-mul-action-Group : type-Group G  type-action-Group  type-action-Group
  equiv-mul-action-Group =
    map-hom-Group G (symmetric-Group set-action-Group) (pr2 X)

  mul-action-Group : type-Group G  type-action-Group  type-action-Group
  mul-action-Group g = map-equiv (equiv-mul-action-Group g)

  mul-action-Group' : type-action-Group  type-Group G  type-action-Group
  mul-action-Group' x g = mul-action-Group g x

  preserves-unit-mul-action-Group : mul-action-Group (unit-Group G) ~ id
  preserves-unit-mul-action-Group =
      ( ap pr1
        ( preserves-unit-hom-Group G
          ( symmetric-Group set-action-Group)
          ( pr2 X)))

  preserves-mul-action-Group :
    (g : type-Group G) (h : type-Group G) (x : type-action-Group) 
    mul-action-Group (mul-Group G g h) x 
    mul-action-Group g (mul-action-Group h x)
  preserves-mul-action-Group g h =
      ( ap pr1
        ( preserves-mul-hom-Group G (symmetric-Group set-action-Group) (pr2 X)))

  transpose-eq-mul-action-Group :
    (g : type-Group G) (x y : type-action-Group) 
    mul-action-Group g x  y  x  mul-action-Group (inv-Group G g) y
  transpose-eq-mul-action-Group g x ._ refl =
    ( inv
      ( ( ap (mul-action-Group' x) (left-inverse-law-mul-Group G g)) 
        ( preserves-unit-mul-action-Group x))) 
    ( preserves-mul-action-Group (inv-Group G g) g x)


Trivial G-sets

Every set gives rise to a G-set by having every point fixed under the action of G.

module _
  {l1 l2 : Level} (G : Group l1) (X : Set l2)

  trivial-action-Group : action-Group G l2
  pr1 trivial-action-Group = X
  pr2 trivial-action-Group = trivial-hom-Group G (symmetric-Group X)

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