(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.                  *)
(* Distributed under the terms of CeCILL-B.                                  *)
From mathcomp
From mathcomp
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(* Group automorphisms and characteristic subgroups.                          *)
(* Unlike morphisms on a group G, which are functions of type gT -> rT, with  *)
(* a canonical structure of dependent type {morphim G >-> rT}, automorphisms  *)
(* are permutations of type {perm gT} contained in Aut G : {set {perm gT}}.   *)
(* This lets us use the finGroupType of {perm gT}. Note also that while       *)
(* morphisms on G are undefined outside G, automorphisms have their support   *)
(* in G, i.e., they are the identity outside G.                               *)
(* Definitions:                                                               *)
(*    Aut G (or [Aut G]) == the automorphism group of G.                      *)
(*             [Aut G]%G == the group structure for Aut G.                    *)
(*            autm AutGa == the morphism on G induced by a, given             *)
(*                          AutGa : a \in Aut G.                              *)
(*       perm_in injf fA == the permutation with support B in induced by f,   *)
(*                          given injf : {in A &, injective f} and            *)
(*                          fA : f @: A \subset A.                            *)
(*           aut injf fG == the automorphism of G induced by the morphism f,  *)
(*                          given injf : 'injm f and fG : f @* G \subset G.   *)
(*    Aut_isom injf sDom == the injective homomorphism that maps Aut G to     *)
(*                          Aut (f @* G), with f : {morphism D >-> rT} and    *)
(*                          given injf: 'injm f and sDom : G \subset D.       *)
(*              conjgm G == the conjugation automorphism on G.                *)
(*             H \char G == H is a characteristic subgroup of G.              *)
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Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
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(* A group automorphism, defined as a permutation on a subset of a     *)
(* finGroupType that respects the morphism law.                        *)
(* Here perm_on is used as a closure rule for the set A.               *)
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Section Automorphism.
Variable gT : finGroupType.
Implicit Type A : {set gT}.
Implicit Types a b : {perm gT}.
Definition Aut A := [set a | perm_on A a & morphic A a].

(* Specifications: *)
Lemma Aut_morphic A a : a \in Aut A -> morphic A a.
Definition autm A a (AutAa : a \in Aut A) := morphm (Aut_morphic AutAa).
(* The morphism that is represented by a given element of Aut A. *)
Lemma autmE A a (AutAa : a \in Aut A) : autm AutAa = a.
Canonical Aut_group := group Aut_group_set.