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improve normal subgroup lemmas #2179

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Jan 3, 2025
Merged

improve normal subgroup lemmas #2179

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improve normal subgroup lemmas
Alizter Jan 3, 2025
4a4c612
Merge branch 'master' into ps/rr/improve_normal_subgroup_lemmas
Alizter Jan 3, 2025
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17 changes: 7 additions & 10 deletions theories/Algebra/Groups/Subgroup.v
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Original file line number Diff line number Diff line change
Expand Up @@ -372,7 +372,7 @@ Coercion normalsubgroup_subgroup : NormalSubgroup >-> Subgroup.
Global Existing Instance normalsubgroup_isnormal.

Definition equiv_symmetric_in_normalsubgroup {G : Group}
(N : NormalSubgroup G)
(N : Subgroup G) `{!IsNormalSubgroup N}
: forall x y, N (x * y) <~> N (y * x).
Proof.
intros x y.
Expand All @@ -381,15 +381,12 @@ Proof.
Defined.

(** Our definiiton of normal subgroup implies the usual definition of invariance under conjugation. *)
Definition isnormal_conjugate {G : Group} (N : NormalSubgroup G) {x y : G}
: N x -> N (y * x * y^).
Definition isnormal_conj {G : Group} (N : Subgroup G)
`{!IsNormalSubgroup N} {x y : G}
: N x <~> N (y * x * y^).
Proof.
intros n.
apply isnormal.
nrefine (transport N (grp_assoc _ _ _)^ _).
nrefine (transport (fun y => N (y * x)) (grp_inv_l _)^ _).
nrefine (transport N (grp_unit_l _)^ _).
exact n.
srefine (equiv_symmetric_in_normalsubgroup N _ _ oE _).
by rewrite grp_inv_V_gg.
Defined.

(** We can show a subgroup is normal if it is invariant under conjugation. *)
Expand Down Expand Up @@ -418,7 +415,7 @@ Definition equiv_isnormal_conjugate `{Funext} {G : Group} (N : Subgroup G)
Proof.
rapply equiv_iff_hprop.
- intros is_normal x y.
exact (isnormal_conjugate (Build_NormalSubgroup G N is_normal)).
rapply isnormal_conj.
- intros is_normal'.
by snrapply Build_IsNormalSubgroup'.
Defined.
Expand Down
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