(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.                  *)
(* Distributed under the terms of CeCILL-B.                                  *)
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(******************************************************************************)
(*  This file regroups constructions and results that are based on the most   *)
(* primitive version of representation theory -- viewing an abelian group as  *)
(* the additive group of a (finite) Z-module. This includes the Gaschutz      *)
(* splitting and transitivity theorem, from which we will later derive the    *)
(* Schur-Zassenhaus theorem and the elementary abelian special case of        *)
(* Maschke's theorem, the coprime abelian centraliser/commutator trivial      *)
(* intersection theorem, which is used to show that p-groups under coprime    *)
(* action factor into special groups, and the construction of the transfer    *)
(* homomorphism and its expansion relative to a cycle, from which we derive   *)
(* the Higman Focal Subgroup and the Burnside Normal Complement theorems.     *)
(*   The definitions and lemmas for the finite Z-module induced by an abelian *)
(* are packaged in an auxiliary FiniteModule submodule: they should not be    *)
(* needed much outside this file, which contains all the results that exploit *)
(* this construction.                                                         *)
(*   FiniteModule defines the Z[N(A)]-module associated with a finite abelian *)
(* abelian group A, given a proof (abelA : abelian A) :                       *)
(*  fmod_of abelA == the type of elements of the module (similar to but       *)
(*                   distinct from [subg A]).                                 *)
(*   fmod abelA x == the injection of x into fmod_of abelA if x \in A, else 0 *)
(*        fmval u == the projection of u : fmod_of abelA onto A               *)
(*         u ^@ x == the action of x \in 'N(A) on u : fmod_of abelA           *)
(* The transfer morphism is be constructed from a morphism f : H >-> rT, and  *)
(* a group G, along with the two assumptions sHG : H \subset G and            *)
(* abfH : abelian (f @* H):                                                   *)
(*  transfer sGH abfH == the function gT -> FiniteModule.fmod_of abfH that    *)
(*                   implements the transfer morphism induced by f on G.      *)
(* The Lemma transfer_indep states that the transfer morphism can be expanded *)
(* using any transversal of the partition HG := rcosets H G of G.             *)
(*   Further, for any g \in G, HG :* <[g]> is also a partition of G (Lemma    *)
(* rcosets_cycle_partition), and for any transversal X of HG :* <[g]> the     *)
(* function r mapping x : gT to rcosets (H :* x) <[g]> is (constructively) a  *)
(* bijection from X to the <[g]>-orbit partition of HG, and Lemma             *)
(* transfer_pcycle_def gives a simplified expansion of the transfer morphism. *)
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Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Module FiniteModule.
Reserved Notation "u ^@ x" (at level 31, left associativity).
Inductive fmod_of (gT : finGroupType) (A : {group gT}) (abelA : abelian A) :=
Fmod x & x \in A.
Bind Scope ring_scope with fmod_of.
Section OneFinMod.
Let f2sub (gT : finGroupType) (A : {group gT}) (abA : abelian A) :=
fun u : fmod_of abA => let : Fmod x Ax := u in Subg Ax : FinGroup.arg_sort _.
Local Coercion f2sub : fmod_of >-> FinGroup.arg_sort.
Variables (gT : finGroupType) (A : {group gT}) (abelA : abelian A).
Local Notation fmodA := (fmod_of abelA).
Implicit Types (x y z : gT) (u v w : fmodA).
Let sub2f (s : [subg A]) := Fmod abelA (valP s).
Definition fmval u := val (f2sub u).
Canonical fmod_subType := [subType for fmval].
Local Notation valA := (@val _ _ fmod_subType) (only parsing).
Definition fmod_eqMixin := Eval hnf in [eqMixin of fmodA by <:].
Canonical fmod_eqType := Eval hnf in EqType fmodA fmod_eqMixin.
Definition fmod_choiceMixin := [choiceMixin of fmodA by <:].
Canonical fmod_choiceType := Eval hnf in ChoiceType fmodA fmod_choiceMixin.
Definition fmod_countMixin := [countMixin of fmodA by <:].
Canonical fmod_countType := Eval hnf in CountType fmodA fmod_countMixin.
Canonical fmod_subCountType := Eval hnf in [subCountType of fmodA].
Definition fmod_finMixin := [finMixin of fmodA by <:].
Canonical fmod_finType := Eval hnf in FinType fmodA fmod_finMixin.
Canonical fmod_subFinType := Eval hnf in [subFinType of fmodA].
Definition fmod x := sub2f (subg A x).
Definition actr u x := if x \in 'N(A) then fmod (fmval u ^ x) else u.
Definition fmod_opp u := sub2f u^-1.
Definition fmod_add u v := sub2f (u * v).
Fact fmod_add0r : left_id (sub2f 1) fmod_add.

Lemma injm_fmod : 'injm fmod.
Notation "u ^@ x" := (actr u x) : ring_scope.