Nontrivial groups

Content created by Egbert Rijke and Fredrik Bakke.

Created on 2023-10-08.
Last modified on 2023-12-12.

module group-theory.nontrivial-groups where
Imports
open import foundation.action-on-identifications-functions
open import foundation.contractible-types
open import foundation.dependent-pair-types
open import foundation.disjunction
open import foundation.embeddings
open import foundation.empty-types
open import foundation.existential-quantification
open import foundation.identity-types
open import foundation.injective-maps
open import foundation.logical-equivalences
open import foundation.negated-equality
open import foundation.negation
open import foundation.propositional-extensionality
open import foundation.propositional-truncations
open import foundation.propositions
open import foundation.sets
open import foundation.unit-type
open import foundation.universe-levels

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

Idea

A group is said to be nontrivial if there exists a nonidentity element.

Definitions

The predicate of being a nontrivial group

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

  is-nontrivial-prop-Group : Prop l1
  is-nontrivial-prop-Group =
    ∃-Prop (type-Group G)  g  unit-Group G  g)

  is-nontrivial-Group : UU l1
  is-nontrivial-Group =
    type-Prop is-nontrivial-prop-Group

  is-prop-is-nontrivial-Group :
    is-prop is-nontrivial-Group
  is-prop-is-nontrivial-Group =
    is-prop-type-Prop is-nontrivial-prop-Group

The predicate of not being the trivial group

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

  is-not-trivial-prop-Group : Prop l1
  is-not-trivial-prop-Group =
    neg-Prop' ((x : type-Group G)  unit-Group G  x)

  is-not-trivial-Group : UU l1
  is-not-trivial-Group =
    type-Prop is-not-trivial-prop-Group

  is-prop-is-not-trivial-Group :
    is-prop is-not-trivial-Group
  is-prop-is-not-trivial-Group =
    is-prop-type-Prop is-not-trivial-prop-Group

Properties

A group is not a trivial group if and only if it satisfies the predicate of not being trivial

Proof: The proposition ¬ (is-trivial-Group G) holds if and only if G is not contractible, which holds if and only if ¬ ((x : G) → 1 = x).

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

  neg-is-trivial-is-not-trivial-Group :
    is-not-trivial-Group G  ¬ (is-trivial-Group G)
  neg-is-trivial-is-not-trivial-Group H p = H  x  eq-is-contr p)

  is-not-trivial-neg-is-trivial-Group :
    ¬ (is-trivial-Group G)  is-not-trivial-Group G
  is-not-trivial-neg-is-trivial-Group H p = H (unit-Group G , p)

The map subgroup-Prop G : Prop → Subgroup G is an embedding for any nontrivial group

Recall that the subgroup subgroup-Prop G P associated to a proposition P was defined in group-theory.subgroups.

Proof: Suppose that G is a nontrivial group and x is a group element such that 1 ≠ x. Then subgroup-Prop G P = subgroup-Prop G Q if and only if x ∈ subgroup-Prop G P ⇔ x ∈ subgroup-Prop G Q, which holds if and only if P ⇔ Q since x is assumed to be a nonidentity element. This shows that subgroup-Prop G : Prop → Subgroup G is an injective map. Since it is an injective maps between sets, it follows that subgroup-Prop G is an embedding.

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

  abstract
    is-emb-subgroup-prop-is-nontrivial-Group :
      is-nontrivial-Group G  is-emb (subgroup-Prop {l2 = l2} G)
    is-emb-subgroup-prop-is-nontrivial-Group H =
      apply-universal-property-trunc-Prop H
        ( is-emb-Prop _)
        ( λ (x , f) 
          is-emb-is-injective
            ( is-set-Subgroup G)
            ( λ {P} {Q} α 
              eq-iff
                ( λ p 
                  map-left-unit-law-disjunction-is-empty-Prop
                    ( Id-Prop (set-Group G) _ _)
                    ( Q)
                    ( f)
                    ( forward-implication
                      ( iff-eq (ap  T  subset-Subgroup G T x) α))
                      ( inr-disjunction-Prop
                        ( Id-Prop (set-Group G) _ _)
                        ( P)
                        ( p))))
                ( λ q 
                  map-left-unit-law-disjunction-is-empty-Prop
                    ( Id-Prop (set-Group G) _ _)
                    ( P)
                    ( f)
                    ( backward-implication
                      ( iff-eq (ap  T  subset-Subgroup G T x) α))
                      ( inr-disjunction-Prop
                        ( Id-Prop (set-Group G) _ _)
                        ( Q)
                        ( q))))))

If the map subgroup-Prop G : Prop lzero → Subgroup l1 G is an embedding, then G is not a trivial group

Proof: Suppose that subgroup-Prop G : Prop lzero → Subgroup l1 G is an embedding, and by way of contradiction suppose that G is trivial. Then it follows that Subgroup l1 G is contractible. Since subgroup-Prop G is assumed to be an embedding, it follows that Prop lzero is contractible. This contradicts the fact that Prop lzero contains the distinct propositions empty-Prop and unit-Prop.

Note: Our handling of universe levels might be too restrictive here.

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

  is-not-trivial-is-emb-subgroup-Prop :
    is-emb (subgroup-Prop {l2 = lzero} G)  is-not-trivial-Group G
  is-not-trivial-is-emb-subgroup-Prop H K =
    backward-implication
      ( iff-eq
        ( is-injective-is-emb H
          { x = empty-Prop}
          { y = unit-Prop}
          ( eq-is-contr (is-contr-subgroup-is-trivial-Group G (_ , K)))))
      ( star)

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