Classical.Structures.Group.IndexAction

Right actions of a group on an index set

This is the Classical.Structures.Group.IndexAction module of the Agda Universal Algebra Library.

A right action of a group G on a set I assigns to each group element a map I → I, contravariantly: acting by x ∙ y is acting by x, then by y.

This is the gadget the wreath product construction consumes: the multiplication of S ≀ G twists the base tuple of the right factor by the action of the left factor's second component:

(s , x) (t , y) =  (s₁ tₓ₁ , … , sₙ tₓₙ , x y).

Coordinate i of the product is s i ∙ t (x i).¹ Associativity of the wreath product multiplication comes from the contravariant law (x ∙ y) i = y (x i).

Three design points.

• The index set is a bare type, acted on up to propositional equality ≡, not a setoid. The intended instances are finite index sets Fin n (the cosets of a finite-index subgroup, enumerated), where ≡ is the right equality; keeping the index side propositional lets tuples I → S be permuted by plain precomposition, with no Func bookkeeping. The group side is a setoid, so the action carries a congruence field.

• No bijectivity field. Invertibility of each act x is a consequence of the action laws (act-invˡ, act-invʳ, act-injective), not an axiom; every group action is by bijections.

• Relation to Classical.Structures.Group.GSet. The library encoding of G-sets presents the coset action as a unary algebra on the coset setoid (one operation per group element); that is the form the library's congruence machinery applies to. The present module is the enumerated counterpart: an action on a bare index set, which is what underlies Sᴵ and the wreath product. The two meet in the coset-action specification below.

Kernel–core correspondence. The module closes with the observation that, for an action satisfying the (pointed) coset-action specification for a subgroup H, the action is faithful precisely when H is core-free.² Both directions are proved outright; nothing here is assumed.

{-# OPTIONS --without-K --exact-split --safe #-}

module Classical.Structures.Group.IndexAction where

open import Agda.Primitive using () renaming ( Set to Type )

-- Imports from the Agda Standard Library ---------------------------------------
open import Data.Product                           using  ( Σ-syntax ; proj₁ ; proj₂ ; _,_ )
open import Level                                  using  ( Level ; _⊔_ )
open import Relation.Binary                        using  ( Setoid )
open import Relation.Binary.PropositionalEquality  using  ( _≡_ ; sym ; trans ; cong )
open import Relation.Unary                         using  ( Pred ; _∈_ ; _⊆_ )

import Relation.Binary.Reasoning.Setoid as SetoidReasoning

-- Imports from the Agda Universal Algebra Library ------------------------------
open import Classical.Structures.Group.Basic        using  ( Group ; module Group-Op )
open import Classical.Structures.Group.Conjugation  using  ( module Conjugate )
open import Classical.Structures.Group.NormalCore   using  ( module Core )
open import Classical.Structures.Group.Subgroups    using  ( IsSubgroup
                                                           ; trivialSubgroup )
open import Setoid.Algebras.Basic                   using  ( 𝕌[_] ; 𝔻[_] )

private variable ι β σ ℓ : Level

The right-action record

RightActionI 𝒢 packages an action of 𝒢 on the index type I: the map act, its congruence in the group argument (the group carrier is a setoid, and ≈-equal elements must act identically), the identity law, and the contravariant compatibility law.

record RightAction (I : Type ι) (𝒢 : Group β σ) : Type (ι ⊔ β ⊔ σ) where
  open Group-Op 𝒢 using ( _∙_ ; ε ; _⁻¹ ; invˡ-law ; invʳ-law )
  open Setoid 𝔻[ 𝒢 .proj₁ ] using ( _≈_ )

  field
    act       : 𝕌[ 𝒢 .proj₁ ] → I → I
    act-cong  : ∀ {x y} → x ≈ y → ∀ i → act x i ≡ act y i
    act-ε     : ∀ i → act ε i ≡ i
    act-∙     : ∀ x y i → act (x ∙ y) i ≡ act y (act x i)

Every group element acts invertibly: acting by x ⁻¹ undoes acting by x on either side, by one compatibility step, one congruence step along the inverse law, and the identity law. Injectivity of each act x follows.

  -- Acting by x, then by x ⁻¹, is the identity.
  act-invˡ : ∀ x i → act (x ⁻¹) (act x i) ≡ i
  act-invˡ x i =
    trans (sym (act-∙ x (x ⁻¹) i)) (trans (act-cong (invʳ-law x) i) (act-ε i))

  -- Acting by x ⁻¹, then by x, is the identity.
  act-invʳ : ∀ x i → act x (act (x ⁻¹) i) ≡ i
  act-invʳ x i =
    trans (sym (act-∙ (x ⁻¹) x i)) (trans (act-cong (invˡ-law x) i) (act-ε i))

  -- Each group element acts injectively on the index set.
  act-injective : ∀ x {i j} → act x i ≡ act x j → i ≡ j
  act-injective x {i} {j} e =
    trans (sym (act-invˡ x i)) (trans (cong (act (x ⁻¹)) e) (act-invˡ x j))

An action is faithful when only the identity class acts as the identity map — the vanishing of the kernel of the induced permutation representation, in pointwise form.

  -- Only elements ≈ ε act as the identity on every index.
  Faithful : Type (ι ⊔ β ⊔ σ)
  Faithful = ∀ {x} → (∀ i → act x i ≡ i) → x ≈ ε

The coset-action specification

The action Kurzweil's construction uses is the action of G on the (right) cosets of a subgroup H, enumerated by an index set.

We specify it precisely as follows:

• Some index plays the role of the coset H itself, its stabilizer is exactly H, in both directions.
• Every index is reachable from it (transitivity).

This ties the abstract action to the pair (G , H) up to isomorphism of G-sets, which is all any consumer needs, while staying indifferent to the way cosets are enumerated.

record IsCosetAction {I : Type ι} {𝒢 : Group β σ}
  (A : RightAction I 𝒢) (H : Pred 𝕌[ 𝒢 .proj₁ ] ℓ) : Type (ι ⊔ β ⊔ σ ⊔ ℓ)
  where
  open RightAction A

  field
    basepoint  : I
    stab-in    : ∀ {g} → g ∈ H → act g basepoint ≡ basepoint
    stab-out   : ∀ {g} → act g basepoint ≡ basepoint → g ∈ H
    reach      : ∀ i → Σ[ g ∈ 𝕌[ 𝒢 .proj₁ ] ] (act g basepoint ≡ i)

The kernel–core correspondence

For a coset action of H, faithfulness is equivalent to core-freeness of H; classically, the kernel of the action of G on the cosets of H is the normal core Core_G(H).

Both directions are proved outright below, against the library's constructive core (the meet of all conjugates, Classical.Structures.Group.NormalCore); "core-free" is the containment of the core in the ≈-class of the identity.

ActionKernel fixes the data once for both directions.

module ActionKernel {I : Type ι} (𝒢@(𝑮 , eqns) : Group β σ)
  (H : Pred 𝕌[ 𝑮 ] ℓ) (H-sg : IsSubgroup 𝒢 H)
  (A : RightAction I 𝒢) (spec : IsCosetAction A H)
  where
  open Setoid 𝔻[ 𝑮 ]   using ( _≈_ ) renaming ( refl to ≈refl ; sym to ≈sym )
  open SetoidReasoning 𝔻[ 𝑮 ]
  open Group-Op 𝒢      using ( _∙_ ; ε ; _⁻¹ ; assoc-law ; idʳ-law ; invˡ-law ; ∙-cong )
  open Conjugate 𝒢     using ( conj-syntax )
  open Core 𝒢 H H-sg   using ( core ; core-mem-conj ; conj-mem-core )
  open RightAction A
  open IsCosetAction spec

One algebraic step is shared by the second direction: multiplying a conjugate g ∙ x ∙ g ⁻¹ by g on the right recovers g ∙ x.

  private
    -- (g ∙ x ∙ g ⁻¹) ∙ g ≈ g ∙ x, by reassociating and cancelling the inverse.
    conj-slide : ∀ g x → (x ^ g) ∙ g ≈ g ∙ x
    conj-slide g x = begin
      g ∙ x ∙ g ⁻¹ ∙ g    ≈⟨ assoc-law (g ∙ x) (g ⁻¹) g ⟩
      g ∙ x ∙ (g ⁻¹ ∙ g)  ≈⟨ ∙-cong ≈refl (invˡ-law g) ⟩
      g ∙ x ∙ ε           ≈⟨ idʳ-law (g ∙ x) ⟩
      g ∙ x               ∎

Core-free implies faithful. If x acts as the identity on every index, then every conjugate g ∙ x ∙ g ⁻¹ stabilizes the basepoint — unfold the conjugate along the action laws and let x disappear — so by the stabilizer specification every conjugate of x lies in H; that puts x in the core, which core-freeness collapses to the identity class.

  coreFree→faithful : core .proj₁ ⊆ trivialSubgroup 𝒢 .proj₁ → Faithful
  coreFree→faithful cf {x} fix = cf (conj-mem-core conj∈H)
    where
    -- Every conjugate of x stabilizes the basepoint, hence lies in H.
    conj∈H : ∀ g → x ^ g ∈ H
    conj∈H g = stab-out (trans (act-∙ (g ∙ x) (g ⁻¹) basepoint)
      (trans  (cong (act (g ⁻¹)) (trans (act-∙ g x basepoint) (fix (act g basepoint))))
              (act-invˡ g basepoint)))

Faithful implies core-free. A member x of the core has all conjugates in H; to see x fixes an arbitrary index i, reach i from the basepoint by some g and slide the action of g ∙ x through the conjugate: the conjugate stabilizes the basepoint, so act x i ≡ i. Faithfulness then collapses x to the identity class.

  faithful→coreFree : Faithful → core .proj₁ ⊆ trivialSubgroup 𝒢 .proj₁
  faithful→coreFree faith {x} x∈core = faith fix
    where
    fix : ∀ i → act x i ≡ i
    fix i = trans (cong (act x) (sym gb≡i))
      (trans  (sym (act-∙ g x basepoint))
        (trans  (act-cong (≈sym (conj-slide g x)) basepoint)
          (trans  (act-∙ (x ^ g) g basepoint)
            (trans  (cong (act g) (stab-in (core-mem-conj x∈core g)))
                    gb≡i))))
      where
      g : 𝕌[ 𝑮 ]
      g = reach i .proj₁

      gb≡i : act g basepoint ≡ i
      gb≡i = reach i .proj₂

---

---

1. arXiv:1205.1927v4 ("the note"), proof of Lemma 3.3; the wreath product itself is Classical.Structures.Group.Wreath. ↩

2. This is the fact ker φ = 1 ⟺ Core_G(H) = 1 that arguments about core-free coset actions, such as the proof of Lemma 3.3 of the note, rely on. ↩