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quotient is trivial iff subgroup is maximal #2197

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Jan 15, 2025
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quotient is trivial iff subgroup is maximal #2197

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26 changes: 26 additions & 0 deletions theories/Algebra/Groups/QuotientGroup.v
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Expand Up @@ -322,3 +322,29 @@ Definition grp_quotient_trivial (G : Group) (N : NormalSubgroup G)
(triv : IsTrivialGroup N)
: G $<~> G / N
:= Build_CatEquiv grp_quotient_map.

(** A quotient by a maximal subgroup is trivial. *)
Global Instance istrivial_grp_quotient_maximal
(G : Group) (N : NormalSubgroup G)
: IsMaximalSubgroup N -> IsTrivialGroup (G / N).
Proof.
intros max x _; revert x.
rapply grp_quotient_ind_hprop.
intros x.
apply qglue, max.
Defined.

(** Conversely, a trivial quotient means the subgroup is maximal. *)
Definition ismaximalsubgroup_istrivial_grp_quotient `{Univalence}
(G : Group) (N : NormalSubgroup G)
: IsTrivialGroup (G / N) -> IsMaximalSubgroup N.
Proof.
intros triv x.
specialize (triv (grp_quotient_map x) tt).
simpl in triv.
apply related_quotient_paths in triv.
2-5: exact _.
apply equiv_subgroup_inverse.
nrapply (subgroup_in_op_l _ _ 1 triv) .
apply subgroup_in_unit.
Defined.
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