(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.                  *)
(* Distributed under the terms of CeCILL-B.                                  *)
From mathcomp
From mathcomp
From mathcomp
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(* Definition of the center of a group and of external central products:      *)
(*           'Z(G) == the center of the group G, i.e., 'C_G(G).               *)
(*   cprod_by isoZ == the finGroupType for the central product of H and K     *)
(*                    with centers identified by the isomorphism gz on 'Z(H); *)
(*                    here isoZ : isom 'Z(H) 'Z(K) gz. Note that the actual   *)
(*                    central product is [set: cprod_by isoZ].                *)
(*    cpairg1 isoZ == the isomorphism from H to cprod_by isoZ, isoZ as above. *)
(*    cpair1g isoZ == the isomorphism from K to cprod_by isoZ, isoZ as above. *)
(*      xcprod H K == the finGroupType for the external central product of H  *)
(*                    and K with identified centers, provided the dynamically *)
(*                    tested condition 'Z(H) \isog 'Z(K) holds.               *)
(*      ncprod H n == the finGroupType for the central product of n copies of *)
(*                    H with their centers identified; [set: ncprod H 0] is   *)
(*                    isomorphic to 'Z(H).                                    *)
(*  xcprodm cf eqf == the morphism induced on cprod_by isoZ, where as above   *)
(*                    isoZ : isom 'Z(H) 'Z(K) gz, by fH : {morphism H >-> rT} *)
(*                    and fK : {morphism K >-> rT}, given both                *)
(*                    cf : fH @* H \subset 'C(fK @* K) and                    *)
(*                    eqf : {in 'Z(H), fH =1 fK \o gz}.                       *)
(*   Following Aschbacher, we only provide external central products with     *)
(* identified centers, as these are well defined provided the local center    *)
(* isomorphism group of one of the subgroups is full. Nevertheless the        *)
(* entire construction could be carried out under the weaker assumption that  *)
(* gz is an isomorphism between subgroups of 'Z(H) and 'Z(K), and even the    *)
(* uniqueness theorem holds under the weaker assumption that gz map 'Z(H) to  *)
(* a characteristic subgroup of 'Z(K) not isomorphic to any other subgroup of *)
(* 'Z(K), a condition that holds for example when K is cyclic, as in the      *)
(* structure theorem for p-groups of symplectic type.                         *)
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Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Section Defs.
Variable gT : finGroupType.
Definition center (A : {set gT}) := 'C_A(A).
Canonical center_group (G : {group gT}) : {group gT} :=
Eval hnf in [group of center G].
End Defs.
Arguments center {gT} A%g.
Notation "''Z' ( A )" := (center A) : group_scope.
Notation "''Z' ( H )" := (center_group H) : Group_scope.

Lemma cyclic_factor_abelian H G :
H \subset 'Z(G) -> cyclic (G / H) -> abelian G.