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trivial and maximal subgroup are normal #2181

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Jan 3, 2025
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trivial and maximal subgroup are normal #2181

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16 changes: 16 additions & 0 deletions theories/Algebra/Groups/Subgroup.v
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Expand Up @@ -423,6 +423,22 @@ Proof.
by snrapply Build_IsNormalSubgroup'.
Defined.

(** The trivial subgroup is a normal subgroup. *)
Global Instance isnormal_trivial_subgroup {G : Group}
: IsNormalSubgroup (trivial_subgroup G).
Proof.
intros x y p; cbn in p |- *.
apply grp_moveL_1V in p.
by apply grp_moveL_V1.
Defined.

(** The maximal subgroup (the group itself) is a normal subgroup. *)
Global Instance isnormal_maximal_subgroup {G : Group}
: IsNormalSubgroup (maximal_subgroup G).
Proof.
intros x y p; exact tt.
Defined.

(** Left and right cosets are equivalent in normal subgroups. *)
Definition equiv_in_cosetL_in_cosetR_normalsubgroup {G : Group}
(N : NormalSubgroup G) (x y : G)
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