Centralizer.v

Require Import Basics Types Truncations.Core.

Require Import HFiber AbelianGroup.


(* Given a group [G], we define the centralizer of an element [g : G] as a subgroup and use this to show that the cyclic subgroup generated by [g] is abelian. *)

Local Open Scope mc_scope.

Local Open Scope mc_mult_scope.


(* First we show that the collection of elements that commute with a fixed element [g] is a subgroup. *)

Definition centralizer {G : Group} (g : G)
  := fun h => g * h = h * g.


Definition centralizer_unit {G : Group} (g : G) : centralizer g mon_unit.

Proof.

  exact (grp_unit_r _ @ (grp_unit_l _)^).

Defined.


Definition centralizer_sgop {G : Group} (g h k : G)
           (p : centralizer g h) (q : centralizer g k)
  : centralizer g (h * k).

Proof.

  refine (grp_assoc _ _ _ @ _).

  refine (ap (fun x => x * k) p @ _).

  refine ((grp_assoc _ _ _)^ @ _).

  refine (ap (fun x => h * x) q @ _).

  apply grp_assoc.

Defined.


Definition centralizer_inverse {G : Group} (g h : G)
           (p : centralizer g h)
  : centralizer g (-h).

Proof.

  unfold centralizer in *.

  symmetry.

  refine ((grp_unit_r _)^ @ _ @ grp_unit_l _).

  refine (ap (fun x => (-h * g * x)) (grp_inv_r h)^ @ _ @ ap (fun x => x * (g * -h)) (grp_inv_l h)).

  refine (grp_assoc _ _ _ @ _ @ (grp_assoc _ _ _)^).

  refine (ap (fun x => x * (-h)) _).

  refine ((grp_assoc _ _ _)^ @ _ @ grp_assoc _ _ _).

  exact (ap (fun x => (-h) * x) p).

Defined.


Global Instance issubgroup_centralizer {G : Group} (g : G)
  : IsSubgroup (centralizer g).

Proof.

  srapply Build_IsSubgroup.

 apply centralizer_unit.

 apply centralizer_sgop.

 apply centralizer_inverse.

Defined.


Definition centralizer_subgroup {G : Group} (g : G)
  := Build_Subgroup G (centralizer g) _.


(* Now we define cyclic subgroups.  We allow any map [Unit -> G] in this definition, because in applications (such as [Z_commutative]) we have no control over the map. *)

Definition cyclic_subgroup_from_unit {G : Group} (gen : Unit -> G) := subgroup_generated (hfiber gen).


(* When we have a particular element [g] of [G], we could choose the predicate to be [fun h => h = g], but to fit into the above definition, we use [unit_name g], which gives the predicate [fun h => hfiber (unit_name g) h]. *)

Definition cyclic_subgroup {G : Group} (g : G) := cyclic_subgroup_from_unit (unit_name g).


(* Any cyclic subgroup is commutative. *)

Global Instance commutative_cyclic_subgroup {G : Group} (gen : Unit -> G)
  : Commutative (@group_sgop (cyclic_subgroup_from_unit gen)).

Proof.

  intros h k.

  destruct h as [h H]; cbn in H.

  destruct k as [k K]; cbn in K.

  strip_truncations.

  (* It's enough to check equality after including into G: *)

apply (equiv_ap_isembedding (subgroup_incl _) _ _)^-1.

  cbn.

  induction H as [h [[] p]| |h1 h2 H1 H2 IHH1 IHH2].

 (* The case when h = g: *)

induction p.

    induction K as [k [[] q]| |k1 k2 K1 K2 IHK1 IHK2].

 (* The case when k = g: *)

induction q.

      reflexivity.

 (* The case when k = mon_unit: *)

apply centralizer_unit.

 (* The case when k = k1 (-k2): *)

srapply (issubgroup_in_op_inv (H:=centralizer (gen tt))); assumption.

 (* The case when h = mon_unit: *)

symmetry; apply centralizer_unit.

 (* The case when h = h1 (-h2): *)

symmetry.

    srapply (issubgroup_in_op_inv (H:=centralizer k)); unfold centralizer; symmetry; assumption.

Defined.


Definition abgroup_cyclic_subgroup {G : Group} (g : G) : AbGroup
  := Build_AbGroup (cyclic_subgroup g) _.

Timings for Centralizer.v