(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. *) (* Distributed under the terms of CeCILL-B. *) From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div. From mathcomp Require Import choice fintype bigop finset fingroup morphism. From mathcomp Require Import quotient action. (******************************************************************************) (* Partial, semidirect, central, and direct products. *) (* ++ Internal products, with A, B : {set gT}, are partial operations : *) (* partial_product A B == A * B if A is a group normalised by the group B, *) (* and the empty set otherwise. *) (* A ><| B == A * B if this is a semi-direct product (i.e., if A *) (* is normalised by B and intersects it trivially). *) (* A \* B == A * B if this is a central product ([A, B] = 1). *) (* A \x B == A * B if this is a direct product. *) (* [complements to K in G] == set of groups H s.t. K * H = G and K :&: H = 1. *) (* [splits G, over K] == [complements to K in G] is not empty. *) (* remgr A B x == the right remainder in B of x mod A, i.e., *) (* some element of (A :* x) :&: B. *) (* divgr A B x == the "division" in B of x by A: for all x, *) (* x = divgr A B x * remgr A B x. *) (* ++ External products : *) (* pairg1, pair1g == the isomorphisms aT1 -> aT1 * aT2, aT2 -> aT1 * aT2. *) (* (aT1 * aT2 has a direct product group structure.) *) (* sdprod_by to == the semidirect product defined by to : groupAction H K. *) (* This is a finGroupType; the actual semidirect product is *) (* the total set [set: sdprod_by to] on that type. *) (* sdpair[12] to == the isomorphisms injecting K and H into *) (* sdprod_by to = sdpair1 to @* K ><| sdpair2 to @* H. *) (* External central products (with identified centers) will be defined later *) (* in file center.v. *) (* ++ Morphisms on product groups: *) (* pprodm nAB fJ fAB == the morphism extending fA and fB on A <*> B when *) (* nAB : B \subset 'N(A), *) (* fJ : {in A & B, morph_act 'J 'J fA fB}, and *) (* fAB : {in A :&: B, fA =1 fB}. *) (* sdprodm defG fJ == the morphism extending fA and fB on G, given *) (* defG : A ><| B = G and *) (* fJ : {in A & B, morph_act 'J 'J fA fB}. *) (* xsdprodm fHKact == the total morphism on sdprod_by to induced by *) (* fH : {morphism H >-> rT}, fK : {morphism K >-> rT}, *) (* with to : groupAction K H, *) (* given fHKact : morph_act to 'J fH fK. *) (* cprodm defG cAB fAB == the morphism extending fA and fB on G, when *) (* defG : A \* B = G, *) (* cAB : fB @* B \subset 'C(fB @* A), *) (* and fAB : {in A :&: B, fA =1 fB}. *) (* dprodm defG cAB == the morphism extending fA and fB on G, when *) (* defG : A \x B = G and *) (* cAB : fA @* B \subset 'C(fA @* A) *) (* mulgm (x, y) == x * y; mulgm is an isomorphism from setX A B to G *) (* iff A \x B = G . *) (******************************************************************************) Set Implicit Arguments. Unset Strict Implicit. Unset Printing Implicit Defensive. Import GroupScope. Section Defs. Variables gT : finGroupType. Implicit Types A B C : {set gT}. Definition partial_product A B := if A == 1 then B else if B == 1 then A else if [&& group_set A, group_set B & B \subset 'N(A)] then A * B else set0. Definition semidirect_product A B := if A :&: B \subset 1%G then partial_product A B else set0. Definition central_product A B := if B \subset 'C(A) then partial_product A B else set0. Definition direct_product A B := if A :&: B \subset 1%G then central_product A B else set0. Definition complements_to_in A B := [set K : {group gT} | A :&: K == 1 & A * K == B]. Definition splits_over B A := complements_to_in A B != set0. (* Product remainder functions -- right variant only. *) Definition remgr A B x := repr (A :* x :&: B). Definition divgr A B x := x * (remgr A B x)^-1. End Defs. Arguments partial_product _ _%g _%g : clear implicits. Arguments semidirect_product _ _%g _%g : clear implicits. Arguments central_product _ _%g _%g : clear implicits. Arguments complements_to_in _ _%g _%g. Arguments splits_over _ _%g _%g. Arguments remgr _ _%g _%g _%g. Arguments divgr _ _%g _%g _%g. Arguments direct_product : clear implicits. Notation pprod := (partial_product _). Notation sdprod := (semidirect_product _). Notation cprod := (central_product _). Notation dprod := (direct_product _). Notation "G ><| H" := (sdprod G H)%g (at level 40, left associativity) : group_scope. Notation "G \* H" := (cprod G H)%g (at level 40, left associativity) : group_scope. Notation "G \x H" := (dprod G H)%g (at level 40, left associativity) : group_scope. Notation "[ 'complements' 'to' A 'in' B ]" := (complements_to_in A B) (at level 0, format "[ 'complements' 'to' A 'in' B ]") : group_scope. Notation "[ 'splits' B , 'over' A ]" := (splits_over B A) (at level 0, format "[ 'splits' B , 'over' A ]") : group_scope. (* Prenex Implicits remgl divgl. *) Prenex Implicits remgr divgr. Section InternalProd. Variable gT : finGroupType. Implicit Types A B C : {set gT}. Implicit Types G H K L M : {group gT}. Local Notation pprod := (partial_product gT). Local Notation sdprod := (semidirect_product gT) (only parsing). Local Notation cprod := (central_product gT) (only parsing). Local Notation dprod := (direct_product gT) (only parsing). Lemma pprod1g : left_id 1 pprod. Proof. by move=> A; rewrite /pprod eqxx. Qed. Lemma pprodg1 : right_id 1 pprod. Proof. by move=> A; rewrite /pprod eqxx; case: eqP. Qed. Variant are_groups A B : Prop := AreGroups K H of A = K & B = H. Lemma group_not0 G : set0 <> G. Proof. by move/setP/(_ 1); rewrite inE group1. Qed. Lemma mulg0 : right_zero (@set0 gT) mulg. Proof. by move=> A; apply/setP=> x; rewrite inE; apply/imset2P=> [[y z]]; rewrite inE. Qed. Lemma mul0g : left_zero (@set0 gT) mulg. Proof. by move=> A; apply/setP=> x; rewrite inE; apply/imset2P=> [[y z]]; rewrite inE. Qed. Lemma pprodP A B G : pprod A B = G -> [/\ are_groups A B, A * B = G & B \subset 'N(A)]. Proof. have Gnot0 := @group_not0 G; rewrite /pprod; do 2?case: eqP => [-> ->| _]. - by rewrite mul1g norms1; split; first exists 1%G G. - by rewrite mulg1 sub1G; split; first exists G 1%G. by case: and3P => // [[gA gB ->]]; split; first exists (Group gA) (Group gB). Qed. Lemma pprodE K H : H \subset 'N(K) -> pprod K H = K * H. Proof. move=> nKH; rewrite /pprod nKH !groupP /=. by do 2?case: eqP => [-> | _]; rewrite ?mulg1 ?mul1g. Qed. Lemma pprodEY K H : H \subset 'N(K) -> pprod K H = K <*> H. Proof. by move=> nKH; rewrite pprodE ?norm_joinEr. Qed. Lemma pprodW A B G : pprod A B = G -> A * B = G. Proof. by case/pprodP. Qed. Lemma pprodWC A B G : pprod A B = G -> B * A = G. Proof. by case/pprodP=> _ <- /normC. Qed. Lemma pprodWY A B G : pprod A B = G -> A <*> B = G. Proof. by case/pprodP=> [[K H -> ->] <- /norm_joinEr]. Qed. Lemma pprodJ A B x : pprod A B :^ x = pprod (A :^ x) (B :^ x). Proof. rewrite /pprod !conjsg_eq1 !group_setJ normJ conjSg -conjsMg. by do 3?case: ifP => // _; apply: conj0g. Qed. (* Properties of the remainders *) Lemma remgrMl K B x y : y \in K -> remgr K B (y * x) = remgr K B x. Proof. by move=> Ky; rewrite {1}/remgr rcosetM rcoset_id. Qed. Lemma remgrP K B x : (remgr K B x \in K :* x :&: B) = (x \in K * B). Proof. set y := _ x; apply/idP/mulsgP=> [|[g b Kg Bb x_gb]]. rewrite inE rcoset_sym mem_rcoset => /andP[Kxy' By]. by exists (x * y^-1) y; rewrite ?mulgKV. by apply: (mem_repr b); rewrite inE rcoset_sym mem_rcoset x_gb mulgK Kg. Qed. Lemma remgr1 K H x : x \in K -> remgr K H x = 1. Proof. by move=> Kx; rewrite /remgr rcoset_id ?repr_group. Qed. Lemma divgr_eq A B x : x = divgr A B x * remgr A B x. Proof. by rewrite mulgKV. Qed. Lemma divgrMl K B x y : x \in K -> divgr K B (x * y) = x * divgr K B y. Proof. by move=> Hx; rewrite /divgr remgrMl ?mulgA. Qed. Lemma divgr_id K H x : x \in K -> divgr K H x = x. Proof. by move=> Kx; rewrite /divgr remgr1 // invg1 mulg1. Qed. Lemma mem_remgr K B x : x \in K * B -> remgr K B x \in B. Proof. by rewrite -remgrP => /setIP[]. Qed. Lemma mem_divgr K B x : x \in K * B -> divgr K B x \in K. Proof. by rewrite -remgrP inE rcoset_sym mem_rcoset => /andP[]. Qed. Section DisjointRem. Variables K H : {group gT}. Hypothesis tiKH : K :&: H = 1. Lemma remgr_id x : x \in H -> remgr K H x = x. Proof. move=> Hx; apply/eqP; rewrite eq_mulgV1 (sameP eqP set1gP) -tiKH inE. rewrite -mem_rcoset groupMr ?groupV // -in_setI remgrP. by apply: subsetP Hx; apply: mulG_subr. Qed. Lemma remgrMid x y : x \in K -> y \in H -> remgr K H (x * y) = y. Proof. by move=> Kx Hy; rewrite remgrMl ?remgr_id. Qed. Lemma divgrMid x y : x \in K -> y \in H -> divgr K H (x * y) = x. Proof. by move=> Kx Hy; rewrite /divgr remgrMid ?mulgK. Qed. End DisjointRem. (* Intersection of a centraliser with a disjoint product. *) Lemma subcent_TImulg K H A : K :&: H = 1 -> A \subset 'N(K) :&: 'N(H) -> 'C_K(A) * 'C_H(A) = 'C_(K * H)(A). Proof. move=> tiKH /subsetIP[nKA nHA]; apply/eqP. rewrite group_modl ?subsetIr // eqEsubset setSI ?mulSg ?subsetIl //=. apply/subsetP=> _ /setIP[/mulsgP[x y Kx Hy ->] cAxy]. rewrite inE cAxy mem_mulg // inE Kx /=. apply/centP=> z Az; apply/commgP/conjg_fixP. move/commgP/conjg_fixP/(congr1 (divgr K H)): (centP cAxy z Az). by rewrite conjMg !divgrMid ?memJ_norm // (subsetP nKA, subsetP nHA). Qed. (* Complements, and splitting. *) Lemma complP H A B : reflect (A :&: H = 1 /\ A * H = B) (H \in [complements to A in B]). Proof. by apply: (iffP setIdP); case; split; apply/eqP. Qed. Lemma splitsP B A : reflect (exists H, H \in [complements to A in B]) [splits B, over A]. Proof. exact: set0Pn. Qed. Lemma complgC H K G : (H \in [complements to K in G]) = (K \in [complements to H in G]). Proof. rewrite !inE setIC; congr (_ && _). by apply/eqP/eqP=> defG; rewrite -(comm_group_setP _) // defG groupP. Qed. Section NormalComplement. Variables K H G : {group gT}. Hypothesis complH_K : H \in [complements to K in G]. Lemma remgrM : K <| G -> {in G &, {morph remgr K H : x y / x * y}}. Proof. case/normalP=> _; case/complP: complH_K => tiKH <- nK_KH x y KHx KHy. rewrite {1}(divgr_eq K H y) mulgA (conjgCV x) {2}(divgr_eq K H x) -2!mulgA. rewrite mulgA remgrMid //; last by rewrite groupMl mem_remgr. by rewrite groupMl !(=^~ mem_conjg, nK_KH, mem_divgr). Qed. Lemma divgrM : H \subset 'C(K) -> {in G &, {morph divgr K H : x y / x * y}}. Proof. move=> cKH; have /complP[_ defG] := complH_K. have nsKG: K <| G by rewrite -defG -cent_joinEr // normalYl cents_norm. move=> x y Gx Gy; rewrite {1}/divgr remgrM // invMg -!mulgA (mulgA y). by congr (_ * _); rewrite -(centsP cKH) ?groupV ?(mem_remgr, mem_divgr, defG). Qed. End NormalComplement. (* Semi-direct product *) Lemma sdprod1g : left_id 1 sdprod. Proof. by move=> A; rewrite /sdprod subsetIl pprod1g. Qed. Lemma sdprodg1 : right_id 1 sdprod. Proof. by move=> A; rewrite /sdprod subsetIr pprodg1. Qed. Lemma sdprodP A B G : A ><| B = G -> [/\ are_groups A B, A * B = G, B \subset 'N(A) & A :&: B = 1]. Proof. rewrite /sdprod; case: ifP => [trAB | _ /group_not0[] //]. case/pprodP=> gAB defG nBA; split=> {defG nBA}//. by case: gAB trAB => H K -> -> /trivgP. Qed. Lemma sdprodE K H : H \subset 'N(K) -> K :&: H = 1 -> K ><| H = K * H. Proof. by move=> nKH tiKH; rewrite /sdprod tiKH subxx pprodE. Qed. Lemma sdprodEY K H : H \subset 'N(K) -> K :&: H = 1 -> K ><| H = K <*> H. Proof. by move=> nKH tiKH; rewrite sdprodE ?norm_joinEr. Qed. Lemma sdprodWpp A B G : A ><| B = G -> pprod A B = G. Proof. by case/sdprodP=> [[K H -> ->] <- /pprodE]. Qed. Lemma sdprodW A B G : A ><| B = G -> A * B = G. Proof. by move/sdprodWpp/pprodW. Qed. Lemma sdprodWC A B G : A ><| B = G -> B * A = G. Proof. by move/sdprodWpp/pprodWC. Qed. Lemma sdprodWY A B G : A ><| B = G -> A <*> B = G. Proof. by move/sdprodWpp/pprodWY. Qed. Lemma sdprodJ A B x : (A ><| B) :^ x = A :^ x ><| B :^ x. Proof. rewrite /sdprod -conjIg sub_conjg conjs1g -pprodJ. by case: ifP => _ //; apply: imset0. Qed. Lemma sdprod_context G K H : K ><| H = G -> [/\ K <| G, H \subset G, K * H = G, H \subset 'N(K) & K :&: H = 1]. Proof. case/sdprodP=> _ <- nKH tiKH. by rewrite /normal mulG_subl mulG_subr mulG_subG normG. Qed. Lemma sdprod_compl G K H : K ><| H = G -> H \in [complements to K in G]. Proof. by case/sdprodP=> _ mulKH _ tiKH; apply/complP. Qed. Lemma sdprod_normal_complP G K H : K <| G -> reflect (K ><| H = G) (K \in [complements to H in G]). Proof. case/andP=> _ nKG; rewrite complgC. apply: (iffP idP); [case/complP=> tiKH mulKH | exact: sdprod_compl]. by rewrite sdprodE ?(subset_trans _ nKG) // -mulKH mulG_subr. Qed. Lemma sdprod_card G A B : A ><| B = G -> (#|A| * #|B|)%N = #|G|. Proof. by case/sdprodP=> [[H K -> ->] <- _ /TI_cardMg]. Qed. Lemma sdprod_isom G A B : A ><| B = G -> {nAB : B \subset 'N(A) | isom B (G / A) (restrm nAB (coset A))}. Proof. case/sdprodP=> [[K H -> ->] <- nKH tiKH]. by exists nKH; rewrite quotientMidl quotient_isom. Qed. Lemma sdprod_isog G A B : A ><| B = G -> B \isog G / A. Proof. by case/sdprod_isom=> nAB; apply: isom_isog. Qed. Lemma sdprod_subr G A B M : A ><| B = G -> M \subset B -> A ><| M = A <*> M. Proof. case/sdprodP=> [[K H -> ->] _ nKH tiKH] sMH. by rewrite sdprodEY ?(subset_trans sMH) //; apply/trivgP; rewrite -tiKH setIS. Qed. Lemma index_sdprod G A B : A ><| B = G -> #|B| = #|G : A|. Proof. case/sdprodP=> [[K H -> ->] <- _ tiHK]. by rewrite indexMg -indexgI setIC tiHK indexg1. Qed. Lemma index_sdprodr G A B M : A ><| B = G -> M \subset B -> #|B : M| = #|G : A <*> M|. Proof. move=> defG; case/sdprodP: defG (defG) => [[K H -> ->] mulKH nKH _] defG sMH. rewrite -!divgS //=; last by rewrite -genM_join gen_subG -mulKH mulgS. by rewrite -(sdprod_card defG) -(sdprod_card (sdprod_subr defG sMH)) divnMl. Qed. Lemma quotient_sdprodr_isom G A B M : A ><| B = G -> M <| B -> {f : {morphism B / M >-> coset_of (A <*> M)} | isom (B / M) (G / (A <*> M)) f & forall L, L \subset B -> f @* (L / M) = A <*> L / (A <*> M)}. Proof. move=> defG nsMH; have [defA defB]: A = <>%G /\ B = <>%G. by have [[K1 H1 -> ->] _ _ _] := sdprodP defG; rewrite /= !genGid. do [rewrite {}defA {}defB; move: {A}<>%G {B}<>%G => K H] in defG nsMH *. have [[nKH /isomP[injKH imKH]] sMH] := (sdprod_isom defG, normal_sub nsMH). have [[nsKG sHG mulKH _ _] nKM] := (sdprod_context defG, subset_trans sMH nKH). have nsKMG: K <*> M <| G. by rewrite -quotientYK // -mulKH -quotientK ?cosetpre_normal ?quotient_normal. have [/= f inj_f im_f] := third_isom (joing_subl K M) nsKG nsKMG. rewrite quotientYidl //= -imKH -(restrm_quotientE nKH sMH) in f inj_f im_f. have /domP[h [_ ker_h _ im_h]]: 'dom (f \o quotm _ nsMH) = H / M. by rewrite ['dom _]morphpre_quotm injmK. have{} im_h L: L \subset H -> h @* (L / M) = K <*> L / (K <*> M). move=> sLH; have [sLG sKKM] := (subset_trans sLH sHG, joing_subl K M). rewrite im_h morphim_comp morphim_quotm [_ @* L]restrm_quotientE ?im_f //. rewrite quotientY ?(normsG sKKM) ?(subset_trans sLG) ?normal_norm //. by rewrite (quotientS1 sKKM) joing1G. exists h => //; apply/isomP; split; last by rewrite im_h //= (sdprodWY defG). by rewrite ker_h injm_comp ?injm_quotm. Qed. Lemma quotient_sdprodr_isog G A B M : A ><| B = G -> M <| B -> B / M \isog G / (A <*> M). Proof. move=> defG; case/sdprodP: defG (defG) => [[K H -> ->] _ _ _] => defG nsMH. by have [h /isom_isog->] := quotient_sdprodr_isom defG nsMH. Qed. Lemma sdprod_modl A B G H : A ><| B = G -> A \subset H -> A ><| (B :&: H) = G :&: H. Proof. case/sdprodP=> {A B} [[A B -> ->]] <- nAB tiAB sAH. rewrite -group_modl ?sdprodE ?subIset ?nAB //. by rewrite setIA tiAB (setIidPl _) ?sub1G. Qed. Lemma sdprod_modr A B G H : A ><| B = G -> B \subset H -> (H :&: A) ><| B = H :&: G. Proof. case/sdprodP=> {A B}[[A B -> ->]] <- nAB tiAB sAH. rewrite -group_modr ?sdprodE ?normsI // ?normsG //. by rewrite -setIA tiAB (setIidPr _) ?sub1G. Qed. Lemma subcent_sdprod B C G A : B ><| C = G -> A \subset 'N(B) :&: 'N(C) -> 'C_B(A) ><| 'C_C(A) = 'C_G(A). Proof. case/sdprodP=> [[H K -> ->] <- nHK tiHK] nHKA {B C G}. rewrite sdprodE ?subcent_TImulg ?normsIG //. by rewrite -setIIl tiHK (setIidPl (sub1G _)). Qed. Lemma sdprod_recl n G K H K1 : #|G| <= n -> K ><| H = G -> K1 \proper K -> H \subset 'N(K1) -> exists G1 : {group gT}, [/\ #|G1| < n, G1 \subset G & K1 ><| H = G1]. Proof. move=> leGn; case/sdprodP=> _ defG nKH tiKH ltK1K nK1H. have tiK1H: K1 :&: H = 1 by apply/trivgP; rewrite -tiKH setSI ?proper_sub. exists (K1 <*> H)%G; rewrite /= -defG sdprodE // norm_joinEr //. rewrite ?mulSg ?proper_sub ?(leq_trans _ leGn) //=. by rewrite -defG ?TI_cardMg // ltn_pmul2r ?proper_card. Qed. Lemma sdprod_recr n G K H H1 : #|G| <= n -> K ><| H = G -> H1 \proper H -> exists G1 : {group gT}, [/\ #|G1| < n, G1 \subset G & K ><| H1 = G1]. Proof. move=> leGn; case/sdprodP=> _ defG nKH tiKH ltH1H. have [sH1H _] := andP ltH1H; have nKH1 := subset_trans sH1H nKH. have tiKH1: K :&: H1 = 1 by apply/trivgP; rewrite -tiKH setIS. exists (K <*> H1)%G; rewrite /= -defG sdprodE // norm_joinEr //. rewrite ?mulgS // ?(leq_trans _ leGn) //=. by rewrite -defG ?TI_cardMg // ltn_pmul2l ?proper_card. Qed. Lemma mem_sdprod G A B x : A ><| B = G -> x \in G -> exists y, exists z, [/\ y \in A, z \in B, x = y * z & {in A & B, forall u t, x = u * t -> u = y /\ t = z}]. Proof. case/sdprodP=> [[K H -> ->{A B}] <- _ tiKH] /mulsgP[y z Ky Hz ->{x}]. exists y; exists z; split=> // u t Ku Ht eqyzut. move: (congr1 (divgr K H) eqyzut) (congr1 (remgr K H) eqyzut). by rewrite !remgrMid // !divgrMid. Qed. (* Central product *) Lemma cprod1g : left_id 1 cprod. Proof. by move=> A; rewrite /cprod cents1 pprod1g. Qed. Lemma cprodg1 : right_id 1 cprod. Proof. by move=> A; rewrite /cprod sub1G pprodg1. Qed. Lemma cprodP A B G : A \* B = G -> [/\ are_groups A B, A * B = G & B \subset 'C(A)]. Proof. by rewrite /cprod; case: ifP => [cAB /pprodP[] | _ /group_not0[]]. Qed. Lemma cprodE G H : H \subset 'C(G) -> G \* H = G * H. Proof. by move=> cGH; rewrite /cprod cGH pprodE ?cents_norm. Qed. Lemma cprodEY G H : H \subset 'C(G) -> G \* H = G <*> H. Proof. by move=> cGH; rewrite cprodE ?cent_joinEr. Qed. Lemma cprodWpp A B G : A \* B = G -> pprod A B = G. Proof. by case/cprodP=> [[K H -> ->] <- /cents_norm/pprodE]. Qed. Lemma cprodW A B G : A \* B = G -> A * B = G. Proof. by move/cprodWpp/pprodW. Qed. Lemma cprodWC A B G : A \* B = G -> B * A = G. Proof. by move/cprodWpp/pprodWC. Qed. Lemma cprodWY A B G : A \* B = G -> A <*> B = G. Proof. by move/cprodWpp/pprodWY. Qed. Lemma cprodJ A B x : (A \* B) :^ x = A :^ x \* B :^ x. Proof. by rewrite /cprod centJ conjSg -pprodJ; case: ifP => _ //; apply: imset0. Qed. Lemma cprod_normal2 A B G : A \* B = G -> A <| G /\ B <| G. Proof. case/cprodP=> [[K H -> ->] <- cKH]; rewrite -cent_joinEr //. by rewrite normalYl normalYr !cents_norm // centsC. Qed. Lemma bigcprodW I (r : seq I) P F G : \big[cprod/1]_(i <- r | P i) F i = G -> \prod_(i <- r | P i) F i = G. Proof. elim/big_rec2: _ G => // i A B _ IH G /cprodP[[_ H _ defB] <- _]. by rewrite (IH H) defB. Qed. Lemma bigcprodWY I (r : seq I) P F G : \big[cprod/1]_(i <- r | P i) F i = G -> << \bigcup_(i <- r | P i) F i >> = G. Proof. elim/big_rec2: _ G => [|i A B _ IH G]; first by rewrite gen0. case/cprodP => [[K H -> defB] <- cKH]. by rewrite -[<<_>>]joing_idr (IH H) ?cent_joinEr -?defB. Qed. Lemma triv_cprod A B : (A \* B == 1) = (A == 1) && (B == 1). Proof. case A1: (A == 1); first by rewrite (eqP A1) cprod1g. apply/eqP=> /cprodP[[G H defA ->]] /eqP. by rewrite defA trivMg -defA A1. Qed. Lemma cprod_ntriv A B : A != 1 -> B != 1 -> A \* B = if [&& group_set A, group_set B & B \subset 'C(A)] then A * B else set0. Proof. move=> A1 B1; rewrite /cprod; case: ifP => cAB; rewrite ?cAB ?andbF //=. by rewrite /pprod -if_neg A1 -if_neg B1 cents_norm. Qed. Lemma trivg0 : (@set0 gT == 1) = false. Proof. by rewrite eqEcard cards0 cards1 andbF. Qed. Lemma group0 : group_set (@set0 gT) = false. Proof. by rewrite /group_set inE. Qed. Lemma cprod0g A : set0 \* A = set0. Proof. by rewrite /cprod centsC sub0set /pprod group0 trivg0 !if_same. Qed. Lemma cprodC : commutative cprod. Proof. rewrite /cprod => A B; case: ifP => cAB; rewrite centsC cAB // /pprod. by rewrite andbCA normC !cents_norm // 1?centsC //; do 2!case: eqP => // ->. Qed. Lemma cprodA : associative cprod. Proof. move=> A B C; case A1: (A == 1); first by rewrite (eqP A1) !cprod1g. case B1: (B == 1); first by rewrite (eqP B1) cprod1g cprodg1. case C1: (C == 1); first by rewrite (eqP C1) !cprodg1. rewrite !(triv_cprod, cprod_ntriv) ?{}A1 ?{}B1 ?{}C1 //. case: isgroupP => [[G ->{A}] | _]; last by rewrite group0. case: (isgroupP B) => [[H ->{B}] | _]; last by rewrite group0. case: (isgroupP C) => [[K ->{C}] | _]; last by rewrite group0 !andbF. case cGH: (H \subset 'C(G)); case cHK: (K \subset 'C(H)); last first. - by rewrite group0. - by rewrite group0 /= mulG_subG cGH andbF. - by rewrite group0 /= centM subsetI cHK !andbF. rewrite /= mulgA mulG_subG centM subsetI cGH cHK andbT -(cent_joinEr cHK). by rewrite -(cent_joinEr cGH) !groupP. Qed. Canonical cprod_law := Monoid.Law cprodA cprod1g cprodg1. Canonical cprod_abelaw := Monoid.ComLaw cprodC. Lemma cprod_modl A B G H : A \* B = G -> A \subset H -> A \* (B :&: H) = G :&: H. Proof. case/cprodP=> [[U V -> -> {A B}]] defG cUV sUH. by rewrite cprodE; [rewrite group_modl ?defG | rewrite subIset ?cUV]. Qed. Lemma cprod_modr A B G H : A \* B = G -> B \subset H -> (H :&: A) \* B = H :&: G. Proof. by rewrite -!(cprodC B) !(setIC H); apply: cprod_modl. Qed. Lemma bigcprodYP (I : finType) (P : pred I) (H : I -> {group gT}) : reflect (forall i j, P i -> P j -> i != j -> H i \subset 'C(H j)) (\big[cprod/1]_(i | P i) H i == (\prod_(i | P i) H i)%G). Proof. apply: (iffP eqP) => [defG i j Pi Pj neq_ij | cHH]. rewrite (bigD1 j) // (bigD1 i) /= ?cprodA in defG; last exact/andP. by case/cprodP: defG => [[K _ /cprodP[//]]]. set Q := P; have sQP: subpred Q P by []; have [n leQn] := ubnP #|Q|. elim: n => // n IHn in (Q) leQn sQP *. have [i Qi | Q0] := pickP Q; last by rewrite !big_pred0. rewrite (cardD1x Qi) add1n ltnS !(bigD1 i Qi) /= in leQn *. rewrite {}IHn {n leQn}// => [|j /andP[/sQP //]]. rewrite bigprodGE cprodEY // gen_subG; apply/bigcupsP=> j /andP[neq_ji Qj]. by rewrite cHH ?sQP. Qed. Lemma bigcprodEY I r (P : pred I) (H : I -> {group gT}) G : abelian G -> (forall i, P i -> H i \subset G) -> \big[cprod/1]_(i <- r | P i) H i = (\prod_(i <- r | P i) H i)%G. Proof. move=> cGG sHG; apply/eqP; rewrite !(big_tnth _ _ r). by apply/bigcprodYP=> i j Pi Pj _; rewrite (sub_abelian_cent2 cGG) ?sHG. Qed. Lemma perm_bigcprod (I : eqType) r1 r2 (A : I -> {set gT}) G x : \big[cprod/1]_(i <- r1) A i = G -> {in r1, forall i, x i \in A i} -> perm_eq r1 r2 -> \prod_(i <- r1) x i = \prod_(i <- r2) x i. Proof. elim: r1 r2 G => [|i r1 IHr] r2 G defG Ax eq_r12. by rewrite perm_sym in eq_r12; rewrite (perm_small_eq _ eq_r12) ?big_nil. have /rot_to[n r3 Dr2]: i \in r2 by rewrite -(perm_mem eq_r12) mem_head. transitivity (\prod_(j <- rot n r2) x j). rewrite Dr2 !big_cons in defG Ax *; have [[_ G1 _ defG1] _ _] := cprodP defG. rewrite (IHr r3 G1) //; first by case/allP/andP: Ax => _ /allP. by rewrite -(perm_cons i) -Dr2 perm_sym perm_rot perm_sym. rewrite -(cat_take_drop n r2) [in LHS]cat_take_drop in eq_r12 *. rewrite (perm_big _ eq_r12) !big_cat /= !(big_nth i) !big_mkord in defG *. have /cprodP[[G1 G2 defG1 defG2] _ /centsP-> //] := defG. rewrite defG2 -(bigcprodW defG2) mem_prodg // => k _; apply: Ax. by rewrite (perm_mem eq_r12) mem_cat orbC mem_nth. rewrite defG1 -(bigcprodW defG1) mem_prodg // => k _; apply: Ax. by rewrite (perm_mem eq_r12) mem_cat mem_nth. Qed. Lemma reindex_bigcprod (I J : finType) (h : J -> I) P (A : I -> {set gT}) G x : {on SimplPred P, bijective h} -> \big[cprod/1]_(i | P i) A i = G -> {in SimplPred P, forall i, x i \in A i} -> \prod_(i | P i) x i = \prod_(j | P (h j)) x (h j). Proof. case=> h1 hK h1K defG Ax; have [e big_e [Ue mem_e] _] := big_enumP P. rewrite -!big_e in defG *; rewrite -(big_map h P x) -[RHS]big_filter filter_map. apply: perm_bigcprod defG _ _ => [i|]; first by rewrite mem_e => /Ax. have [r _ [Ur /= mem_r] _] := big_enumP; apply: uniq_perm Ue _ _ => [|i]. by rewrite map_inj_in_uniq // => i j; rewrite !mem_r ; apply: (can_in_inj hK). rewrite mem_e; apply/idP/mapP=> [Pi|[j r_j ->]]; last by rewrite -mem_r. by exists (h1 i); rewrite ?mem_r h1K. Qed. (* Direct product *) Lemma dprod1g : left_id 1 dprod. Proof. by move=> A; rewrite /dprod subsetIl cprod1g. Qed. Lemma dprodg1 : right_id 1 dprod. Proof. by move=> A; rewrite /dprod subsetIr cprodg1. Qed. Lemma dprodP A B G : A \x B = G -> [/\ are_groups A B, A * B = G, B \subset 'C(A) & A :&: B = 1]. Proof. rewrite /dprod; case: ifP => trAB; last by case/group_not0. by case/cprodP=> gAB; split=> //; case: gAB trAB => ? ? -> -> /trivgP. Qed. Lemma dprodE G H : H \subset 'C(G) -> G :&: H = 1 -> G \x H = G * H. Proof. by move=> cGH trGH; rewrite /dprod trGH sub1G cprodE. Qed. Lemma dprodEY G H : H \subset 'C(G) -> G :&: H = 1 -> G \x H = G <*> H. Proof. by move=> cGH trGH; rewrite /dprod trGH subxx cprodEY. Qed. Lemma dprodEcp A B : A :&: B = 1 -> A \x B = A \* B. Proof. by move=> trAB; rewrite /dprod trAB subxx. Qed. Lemma dprodEsd A B : B \subset 'C(A) -> A \x B = A ><| B. Proof. by rewrite /dprod /cprod => ->. Qed. Lemma dprodWcp A B G : A \x B = G -> A \* B = G. Proof. by move=> defG; have [_ _ _ /dprodEcp <-] := dprodP defG. Qed. Lemma dprodWsd A B G : A \x B = G -> A ><| B = G. Proof. by move=> defG; have [_ _ /dprodEsd <-] := dprodP defG. Qed. Lemma dprodW A B G : A \x B = G -> A * B = G. Proof. by move/dprodWsd/sdprodW. Qed. Lemma dprodWC A B G : A \x B = G -> B * A = G. Proof. by move/dprodWsd/sdprodWC. Qed. Lemma dprodWY A B G : A \x B = G -> A <*> B = G. Proof. by move/dprodWsd/sdprodWY. Qed. Lemma cprod_card_dprod G A B : A \* B = G -> #|A| * #|B| <= #|G| -> A \x B = G. Proof. by case/cprodP=> [[K H -> ->] <- cKH] /cardMg_TI; apply: dprodE. Qed. Lemma dprodJ A B x : (A \x B) :^ x = A :^ x \x B :^ x. Proof. rewrite /dprod -conjIg sub_conjg conjs1g -cprodJ. by case: ifP => _ //; apply: imset0. Qed. Lemma dprod_normal2 A B G : A \x B = G -> A <| G /\ B <| G. Proof. by move/dprodWcp/cprod_normal2. Qed. Lemma dprodYP K H : reflect (K \x H = K <*> H) (H \subset 'C(K) :\: K^#). Proof. rewrite subsetD -setI_eq0 setIDA setD_eq0 setIC subG1 /=. by apply: (iffP andP) => [[cKH /eqP/dprodEY->] | /dprodP[_ _ -> ->]]. Qed. Lemma dprodC : commutative dprod. Proof. by move=> A B; rewrite /dprod setIC cprodC. Qed. Lemma dprodWsdC A B G : A \x B = G -> B ><| A = G. Proof. by rewrite dprodC => /dprodWsd. Qed. Lemma dprodA : associative dprod. Proof. move=> A B C; case A1: (A == 1); first by rewrite (eqP A1) !dprod1g. case B1: (B == 1); first by rewrite (eqP B1) dprod1g dprodg1. case C1: (C == 1); first by rewrite (eqP C1) !dprodg1. rewrite /dprod (fun_if (cprod A)) (fun_if (cprod^~ C)) -cprodA. rewrite -(cprodC set0) !cprod0g cprod_ntriv ?B1 ?{}C1 //. case: and3P B1 => [[] | _ _]; last by rewrite cprodC cprod0g !if_same. case/isgroupP=> H ->; case/isgroupP=> K -> {B C}; move/cent_joinEr=> eHK H1. rewrite cprod_ntriv ?trivMg ?{}A1 ?{}H1 // mulG_subG. case: and4P => [[] | _]; last by rewrite !if_same. case/isgroupP=> G ->{A} _ cGH _; rewrite cprodEY // -eHK. case trGH: (G :&: H \subset _); case trHK: (H :&: K \subset _); last first. - by rewrite !if_same. - rewrite if_same; case: ifP => // trG_HK; case/negP: trGH. by apply: subset_trans trG_HK; rewrite setIS ?joing_subl. - rewrite if_same; case: ifP => // trGH_K; case/negP: trHK. by apply: subset_trans trGH_K; rewrite setSI ?joing_subr. do 2![case: ifP] => // trGH_K trG_HK; [case/negP: trGH_K | case/negP: trG_HK]. apply: subset_trans trHK; rewrite subsetI subsetIr -{2}(mulg1 H) -mulGS. rewrite setIC group_modl ?joing_subr //= cent_joinEr // -eHK. by rewrite -group_modr ?joing_subl //= setIC -(normC (sub1G _)) mulSg. apply: subset_trans trGH; rewrite subsetI subsetIl -{2}(mul1g H) -mulSG. rewrite setIC group_modr ?joing_subl //= eHK -(cent_joinEr cGH). by rewrite -group_modl ?joing_subr //= setIC (normC (sub1G _)) mulgS. Qed. Canonical dprod_law := Monoid.Law dprodA dprod1g dprodg1. Canonical dprod_abelaw := Monoid.ComLaw dprodC. Lemma bigdprodWcp I (r : seq I) P F G : \big[dprod/1]_(i <- r | P i) F i = G -> \big[cprod/1]_(i <- r | P i) F i = G. Proof. elim/big_rec2: _ G => // i A B _ IH G /dprodP[[K H -> defB] <- cKH _]. by rewrite (IH H) // cprodE -defB. Qed. Lemma bigdprodW I (r : seq I) P F G : \big[dprod/1]_(i <- r | P i) F i = G -> \prod_(i <- r | P i) F i = G. Proof. by move/bigdprodWcp; apply: bigcprodW. Qed. Lemma bigdprodWY I (r : seq I) P F G : \big[dprod/1]_(i <- r | P i) F i = G -> << \bigcup_(i <- r | P i) F i >> = G. Proof. by move/bigdprodWcp; apply: bigcprodWY. Qed. Lemma bigdprodYP (I : finType) (P : pred I) (F : I -> {group gT}) : reflect (forall i, P i -> (\prod_(j | P j && (j != i)) F j)%G \subset 'C(F i) :\: (F i)^#) (\big[dprod/1]_(i | P i) F i == (\prod_(i | P i) F i)%G). Proof. apply: (iffP eqP) => [defG i Pi | dxG]. rewrite !(bigD1 i Pi) /= in defG; have [[_ G' _ defG'] _ _ _] := dprodP defG. by apply/dprodYP; rewrite -defG defG' bigprodGE (bigdprodWY defG'). set Q := P; have sQP: subpred Q P by []; have [n leQn] := ubnP #|Q|. elim: n => // n IHn in (Q) leQn sQP *. have [i Qi | Q0] := pickP Q; last by rewrite !big_pred0. rewrite (cardD1x Qi) add1n ltnS !(bigD1 i Qi) /= in leQn *. rewrite {}IHn {n leQn}// => [|j /andP[/sQP //]]. apply/dprodYP; apply: subset_trans (dxG i (sQP i Qi)); rewrite !bigprodGE. by apply: genS; apply/bigcupsP=> j /andP[Qj ne_ji]; rewrite (bigcup_max j) ?sQP. Qed. Lemma dprod_modl A B G H : A \x B = G -> A \subset H -> A \x (B :&: H) = G :&: H. Proof. case/dprodP=> [[U V -> -> {A B}]] defG cUV trUV sUH. rewrite dprodEcp; first by apply: cprod_modl; rewrite ?cprodE. by rewrite setIA trUV (setIidPl _) ?sub1G. Qed. Lemma dprod_modr A B G H : A \x B = G -> B \subset H -> (H :&: A) \x B = H :&: G. Proof. by rewrite -!(dprodC B) !(setIC H); apply: dprod_modl. Qed. Lemma subcent_dprod B C G A : B \x C = G -> A \subset 'N(B) :&: 'N(C) -> 'C_B(A) \x 'C_C(A) = 'C_G(A). Proof. move=> defG; have [_ _ cBC _] := dprodP defG; move: defG. by rewrite !dprodEsd 1?(centSS _ _ cBC) ?subsetIl //; apply: subcent_sdprod. Qed. Lemma dprod_card A B G : A \x B = G -> (#|A| * #|B|)%N = #|G|. Proof. by case/dprodP=> [[H K -> ->] <- _]; move/TI_cardMg. Qed. Lemma bigdprod_card I r (P : pred I) E G : \big[dprod/1]_(i <- r | P i) E i = G -> (\prod_(i <- r | P i) #|E i|)%N = #|G|. Proof. elim/big_rec2: _ G => [G <- | i A B _ IH G defG]; first by rewrite cards1. have [[_ H _ defH] _ _ _] := dprodP defG. by rewrite -(dprod_card defG) (IH H) defH. Qed. Lemma bigcprod_card_dprod I r (P : pred I) (A : I -> {set gT}) G : \big[cprod/1]_(i <- r | P i) A i = G -> \prod_(i <- r | P i) #|A i| <= #|G| -> \big[dprod/1]_(i <- r | P i) A i = G. Proof. elim: r G => [|i r IHr]; rewrite !(big_nil, big_cons) //; case: ifP => _ // G. case/cprodP=> [[K H -> defH]]; rewrite defH => <- cKH leKH_G. have /implyP := leq_trans leKH_G (dvdn_leq _ (dvdn_cardMg K H)). rewrite muln_gt0 leq_pmul2l !cardG_gt0 //= => /(IHr H defH){}defH. by rewrite defH dprodE // cardMg_TI // -(bigdprod_card defH). Qed. Lemma bigcprod_coprime_dprod (I : finType) (P : pred I) (A : I -> {set gT}) G : \big[cprod/1]_(i | P i) A i = G -> (forall i j, P i -> P j -> i != j -> coprime #|A i| #|A j|) -> \big[dprod/1]_(i | P i) A i = G. Proof. move=> defG coA; set Q := P in defG *; have sQP: subpred Q P by []. have [m leQm] := ubnP #|Q|; elim: m => // m IHm in (Q) leQm G defG sQP *. have [i Qi | Q0] := pickP Q; last by rewrite !big_pred0 in defG *. move: defG; rewrite !(bigD1 i Qi) /= => /cprodP[[Hi Gi defAi defGi] <-]. rewrite defAi defGi => cHGi. have{} defGi: \big[dprod/1]_(j | Q j && (j != i)) A j = Gi. by apply: IHm => [||j /andP[/sQP]] //; rewrite (cardD1x Qi) in leQm. rewrite defGi dprodE // coprime_TIg // -defAi -(bigdprod_card defGi). elim/big_rec: _ => [|j n /andP[neq_ji Qj] IHn]; first exact: coprimen1. by rewrite coprimeMr coprime_sym coA ?sQP. Qed. Lemma mem_dprod G A B x : A \x B = G -> x \in G -> exists y, exists z, [/\ y \in A, z \in B, x = y * z & {in A & B, forall u t, x = u * t -> u = y /\ t = z}]. Proof. move=> defG; have [_ _ cBA _] := dprodP defG. by apply: mem_sdprod; rewrite -dprodEsd. Qed. Lemma mem_bigdprod (I : finType) (P : pred I) F G x : \big[dprod/1]_(i | P i) F i = G -> x \in G -> exists c, [/\ forall i, P i -> c i \in F i, x = \prod_(i | P i) c i & forall e, (forall i, P i -> e i \in F i) -> x = \prod_(i | P i) e i -> forall i, P i -> e i = c i]. Proof. move=> defG; rewrite -(bigdprodW defG) => /prodsgP[c Fc ->]. have [r big_r [_ mem_r] _] := big_enumP P. exists c; split=> // e Fe eq_ce i Pi; rewrite -!{}big_r in defG eq_ce. have{Pi}: i \in r by rewrite mem_r. have{mem_r}: all P r by apply/allP=> j; rewrite mem_r. elim: r G defG eq_ce => // j r IHr G. rewrite !big_cons inE /= => /dprodP[[K H defK defH] _ _]. rewrite defK defH => tiFjH eq_ce /andP[Pj Pr]. suffices{i IHr} eq_cej: c j = e j. case/predU1P=> [-> //|]; apply: IHr defH _ Pr. by apply: (mulgI (c j)); rewrite eq_ce eq_cej. rewrite !(big_nth j) !big_mkord in defH eq_ce. move/(congr1 (divgr K H)): eq_ce; move/bigdprodW: defH => defH. move/(all_nthP j) in Pr. by rewrite !divgrMid // -?defK -?defH ?mem_prodg // => *; rewrite ?Fc ?Fe ?Pr. Qed. End InternalProd. Arguments complP {gT H A B}. Arguments splitsP {gT B A}. Arguments sdprod_normal_complP {gT G K H}. Arguments dprodYP {gT K H}. Arguments bigdprodYP {gT I P F}. Section MorphimInternalProd. Variables (gT rT : finGroupType) (D : {group gT}) (f : {morphism D >-> rT}). Section OneProd. Variables G H K : {group gT}. Hypothesis sGD : G \subset D. Lemma morphim_pprod : pprod K H = G -> pprod (f @* K) (f @* H) = f @* G. Proof. case/pprodP=> _ defG mKH; rewrite pprodE ?morphim_norms //. by rewrite -morphimMl ?(subset_trans _ sGD) -?defG // mulG_subl. Qed. Lemma morphim_coprime_sdprod : K ><| H = G -> coprime #|K| #|H| -> f @* K ><| f @* H = f @* G. Proof. rewrite /sdprod => defG coHK; move: defG. by rewrite !coprime_TIg ?coprime_morph // !subxx; apply: morphim_pprod. Qed. Lemma injm_sdprod : 'injm f -> K ><| H = G -> f @* K ><| f @* H = f @* G. Proof. move=> inj_f; case/sdprodP=> _ defG nKH tiKH. by rewrite /sdprod -injmI // tiKH morphim1 subxx morphim_pprod // pprodE. Qed. Lemma morphim_cprod : K \* H = G -> f @* K \* f @* H = f @* G. Proof. case/cprodP=> _ defG cKH; rewrite /cprod morphim_cents // morphim_pprod //. by rewrite pprodE // cents_norm // centsC. Qed. Lemma injm_dprod : 'injm f -> K \x H = G -> f @* K \x f @* H = f @* G. Proof. move=> inj_f; case/dprodP=> _ defG cHK tiKH. by rewrite /dprod -injmI // tiKH morphim1 subxx morphim_cprod // cprodE. Qed. Lemma morphim_coprime_dprod : K \x H = G -> coprime #|K| #|H| -> f @* K \x f @* H = f @* G. Proof. rewrite /dprod => defG coHK; move: defG. by rewrite !coprime_TIg ?coprime_morph // !subxx; apply: morphim_cprod. Qed. End OneProd. Implicit Type G : {group gT}. Lemma morphim_bigcprod I r (P : pred I) (H : I -> {group gT}) G : G \subset D -> \big[cprod/1]_(i <- r | P i) H i = G -> \big[cprod/1]_(i <- r | P i) f @* H i = f @* G. Proof. elim/big_rec2: _ G => [|i fB B Pi def_fB] G sGD defG. by rewrite -defG morphim1. case/cprodP: defG (defG) => [[Hi Gi -> defB] _ _]; rewrite defB => defG. rewrite (def_fB Gi) //; first exact: morphim_cprod. by apply: subset_trans sGD; case/cprod_normal2: defG => _ /andP[]. Qed. Lemma injm_bigdprod I r (P : pred I) (H : I -> {group gT}) G : G \subset D -> 'injm f -> \big[dprod/1]_(i <- r | P i) H i = G -> \big[dprod/1]_(i <- r | P i) f @* H i = f @* G. Proof. move=> sGD injf; elim/big_rec2: _ G sGD => [|i fB B Pi def_fB] G sGD defG. by rewrite -defG morphim1. case/dprodP: defG (defG) => [[Hi Gi -> defB] _ _ _]; rewrite defB => defG. rewrite (def_fB Gi) //; first exact: injm_dprod. by apply: subset_trans sGD; case/dprod_normal2: defG => _ /andP[]. Qed. Lemma morphim_coprime_bigdprod (I : finType) P (H : I -> {group gT}) G : G \subset D -> \big[dprod/1]_(i | P i) H i = G -> (forall i j, P i -> P j -> i != j -> coprime #|H i| #|H j|) -> \big[dprod/1]_(i | P i) f @* H i = f @* G. Proof. move=> sGD /bigdprodWcp defG coH; have def_fG := morphim_bigcprod sGD defG. by apply: bigcprod_coprime_dprod => // i j *; rewrite coprime_morph ?coH. Qed. End MorphimInternalProd. Section QuotientInternalProd. Variables (gT : finGroupType) (G K H M : {group gT}). Hypothesis nMG: G \subset 'N(M). Lemma quotient_pprod : pprod K H = G -> pprod (K / M) (H / M) = G / M. Proof. exact: morphim_pprod. Qed. Lemma quotient_coprime_sdprod : K ><| H = G -> coprime #|K| #|H| -> (K / M) ><| (H / M) = G / M. Proof. exact: morphim_coprime_sdprod. Qed. Lemma quotient_cprod : K \* H = G -> (K / M) \* (H / M) = G / M. Proof. exact: morphim_cprod. Qed. Lemma quotient_coprime_dprod : K \x H = G -> coprime #|K| #|H| -> (K / M) \x (H / M) = G / M. Proof. exact: morphim_coprime_dprod. Qed. End QuotientInternalProd. Section ExternalDirProd. Variables gT1 gT2 : finGroupType. Definition extprod_mulg (x y : gT1 * gT2) := (x.1 * y.1, x.2 * y.2). Definition extprod_invg (x : gT1 * gT2) := (x.1^-1, x.2^-1). Lemma extprod_mul1g : left_id (1, 1) extprod_mulg. Proof. by case=> x1 x2; congr (_, _); apply: mul1g. Qed. Lemma extprod_mulVg : left_inverse (1, 1) extprod_invg extprod_mulg. Proof. by move=> x; congr (_, _); apply: mulVg. Qed. Lemma extprod_mulgA : associative extprod_mulg. Proof. by move=> x y z; congr (_, _); apply: mulgA. Qed. Definition extprod_groupMixin := Eval hnf in FinGroup.Mixin extprod_mulgA extprod_mul1g extprod_mulVg. Canonical extprod_baseFinGroupType := Eval hnf in BaseFinGroupType (gT1 * gT2) extprod_groupMixin. Canonical prod_group := FinGroupType extprod_mulVg. Lemma group_setX (H1 : {group gT1}) (H2 : {group gT2}) : group_set (setX H1 H2). Proof. apply/group_setP; split; first by rewrite !inE !group1. by case=> [x1 x2] [y1 y2] /[!inE] /andP[Hx1 Hx2] /andP[Hy1 Hy2] /[!groupM]. Qed. Canonical setX_group H1 H2 := Group (group_setX H1 H2). Definition pairg1 x : gT1 * gT2 := (x, 1). Definition pair1g x : gT1 * gT2 := (1, x). Lemma pairg1_morphM : {morph pairg1 : x y / x * y}. Proof. by move=> x y /=; rewrite {2}/mulg /= /extprod_mulg /= mul1g. Qed. Canonical pairg1_morphism := @Morphism _ _ setT _ (in2W pairg1_morphM). Lemma pair1g_morphM : {morph pair1g : x y / x * y}. Proof. by move=> x y /=; rewrite {2}/mulg /= /extprod_mulg /= mul1g. Qed. Canonical pair1g_morphism := @Morphism _ _ setT _ (in2W pair1g_morphM). Lemma fst_morphM : {morph (@fst gT1 gT2) : x y / x * y}. Proof. by move=> x y. Qed. Lemma snd_morphM : {morph (@snd gT1 gT2) : x y / x * y}. Proof. by move=> x y. Qed. Canonical fst_morphism := @Morphism _ _ setT _ (in2W fst_morphM). Canonical snd_morphism := @Morphism _ _ setT _ (in2W snd_morphM). Lemma injm_pair1g : 'injm pair1g. Proof. by apply/subsetP=> x /morphpreP[_ /set1P[->]]; apply: set11. Qed. Lemma injm_pairg1 : 'injm pairg1. Proof. by apply/subsetP=> x /morphpreP[_ /set1P[->]]; apply: set11. Qed. Lemma morphim_pairg1 (H1 : {set gT1}) : pairg1 @* H1 = setX H1 1. Proof. by rewrite -imset2_pair imset2_set1r morphimEsub ?subsetT. Qed. Lemma morphim_pair1g (H2 : {set gT2}) : pair1g @* H2 = setX 1 H2. Proof. by rewrite -imset2_pair imset2_set1l morphimEsub ?subsetT. Qed. Lemma morphim_fstX (H1: {set gT1}) (H2 : {group gT2}) : [morphism of fun x => x.1] @* setX H1 H2 = H1. Proof. apply/eqP; rewrite eqEsubset morphimE setTI /=. apply/andP; split; apply/subsetP=> x. by case/imsetP=> x0 /[1!inE] /andP[Hx1 _] ->. move=> Hx1; apply/imsetP; exists (x, 1); last by trivial. by rewrite in_setX Hx1 /=. Qed. Lemma morphim_sndX (H1: {group gT1}) (H2 : {set gT2}) : [morphism of fun x => x.2] @* setX H1 H2 = H2. Proof. apply/eqP; rewrite eqEsubset morphimE setTI /=. apply/andP; split; apply/subsetP=> x. by case/imsetP=> x0 /[1!inE] /andP[_ Hx2] ->. move=> Hx2; apply/imsetP; exists (1, x); last by []. by rewrite in_setX Hx2 andbT. Qed. Lemma setX_prod (H1 : {set gT1}) (H2 : {set gT2}) : setX H1 1 * setX 1 H2 = setX H1 H2. Proof. apply/setP=> [[x y]]; rewrite !inE /=. apply/imset2P/andP=> [[[x1 u1] [v1 y1]] | [Hx Hy]]. rewrite !inE /= => /andP[Hx1 /eqP->] /andP[/eqP-> Hx] [-> ->]. by rewrite mulg1 mul1g. exists (x, 1 : gT2) (1 : gT1, y); rewrite ?inE ?Hx ?eqxx //. by rewrite /mulg /= /extprod_mulg /= mulg1 mul1g. Qed. Lemma setX_dprod (H1 : {group gT1}) (H2 : {group gT2}) : setX H1 1 \x setX 1 H2 = setX H1 H2. Proof. rewrite dprodE ?setX_prod //. apply/centsP=> [[x u]] /[!inE]/= /andP[/eqP-> _] [v y]. by rewrite !inE /= => /andP[_ /eqP->]; congr (_, _); rewrite ?mul1g ?mulg1. apply/trivgP; apply/subsetP=> [[x y]]; rewrite !inE /= -!andbA. by case/and4P=> _ /eqP-> /eqP->; rewrite eqxx. Qed. Lemma isog_setX1 (H1 : {group gT1}) : isog H1 (setX H1 1). Proof. apply/isogP; exists [morphism of restrm (subsetT H1) pairg1]. by rewrite injm_restrm ?injm_pairg1. by rewrite morphim_restrm morphim_pairg1 setIid. Qed. Lemma isog_set1X (H2 : {group gT2}) : isog H2 (setX 1 H2). Proof. apply/isogP; exists [morphism of restrm (subsetT H2) pair1g]. by rewrite injm_restrm ?injm_pair1g. by rewrite morphim_restrm morphim_pair1g setIid. Qed. Lemma setX_gen (H1 : {set gT1}) (H2 : {set gT2}) : 1 \in H1 -> 1 \in H2 -> <> = setX <

> <

>. Proof. move=> H1_1 H2_1; apply/eqP. rewrite eqEsubset gen_subG setXS ?subset_gen //. rewrite -setX_prod -morphim_pair1g -morphim_pairg1 !morphim_gen ?subsetT //. by rewrite morphim_pair1g morphim_pairg1 mul_subG // genS // setXS ?sub1set. Qed. End ExternalDirProd. Section ExternalSDirProd. Variables (aT rT : finGroupType) (D : {group aT}) (R : {group rT}). (* The pair (a, x) denotes the product sdpair2 a * sdpair1 x *) Inductive sdprod_by (to : groupAction D R) : predArgType := SdPair (ax : aT * rT) of ax \in setX D R. Coercion pair_of_sd to (u : sdprod_by to) := let: SdPair ax _ := u in ax. Variable to : groupAction D R. Notation sdT := (sdprod_by to). Notation sdval := (@pair_of_sd to). Canonical sdprod_subType := Eval hnf in [subType for sdval]. Definition sdprod_eqMixin := Eval hnf in [eqMixin of sdT by <:]. Canonical sdprod_eqType := Eval hnf in EqType sdT sdprod_eqMixin. Definition sdprod_choiceMixin := [choiceMixin of sdT by <:]. Canonical sdprod_choiceType := ChoiceType sdT sdprod_choiceMixin. Definition sdprod_countMixin := [countMixin of sdT by <:]. Canonical sdprod_countType := CountType sdT sdprod_countMixin. Canonical sdprod_subCountType := Eval hnf in [subCountType of sdT]. Definition sdprod_finMixin := [finMixin of sdT by <:]. Canonical sdprod_finType := FinType sdT sdprod_finMixin. Canonical sdprod_subFinType := Eval hnf in [subFinType of sdT]. Definition sdprod_one := SdPair to (group1 _). Lemma sdprod_inv_proof (u : sdT) : (u.1^-1, to u.2^-1 u.1^-1) \in setX D R. Proof. by case: u => [[a x]] /= /setXP[Da Rx]; rewrite inE gact_stable !groupV ?Da. Qed. Definition sdprod_inv u := SdPair to (sdprod_inv_proof u). Lemma sdprod_mul_proof (u v : sdT) : (u.1 * v.1, to u.2 v.1 * v.2) \in setX D R. Proof. case: u v => [[a x] /= /setXP[Da Rx]] [[b y] /= /setXP[Db Ry]]. by rewrite inE !groupM //= gact_stable. Qed. Definition sdprod_mul u v := SdPair to (sdprod_mul_proof u v). Lemma sdprod_mul1g : left_id sdprod_one sdprod_mul. Proof. move=> u; apply: val_inj; case: u => [[a x] /=]; case/setXP=> Da _. by rewrite gact1 // !mul1g. Qed. Lemma sdprod_mulVg : left_inverse sdprod_one sdprod_inv sdprod_mul. Proof. move=> u; apply: val_inj; case: u => [[a x] /=]; case/setXP=> Da _. by rewrite actKVin ?mulVg. Qed. Lemma sdprod_mulgA : associative sdprod_mul. Proof. move=> u v w; apply: val_inj; case: u => [[a x]] /=; case/setXP=> Da Rx. case: v w => [[b y]] /=; case/setXP=> Db Ry [[c z]] /=; case/setXP=> Dc Rz. by rewrite !(actMin to) // gactM ?gact_stable // !mulgA. Qed. Canonical sdprod_groupMixin := FinGroup.Mixin sdprod_mulgA sdprod_mul1g sdprod_mulVg. Canonical sdprod_baseFinGroupType := Eval hnf in BaseFinGroupType sdT sdprod_groupMixin. Canonical sdprod_groupType := FinGroupType sdprod_mulVg. Definition sdpair1 x := insubd sdprod_one (1, x) : sdT. Definition sdpair2 a := insubd sdprod_one (a, 1) : sdT. Lemma sdpair1_morphM : {in R &, {morph sdpair1 : x y / x * y}}. Proof. move=> x y Rx Ry; apply: val_inj. by rewrite /= !val_insubd !inE !group1 !groupM ?Rx ?Ry //= mulg1 act1. Qed. Lemma sdpair2_morphM : {in D &, {morph sdpair2 : a b / a * b}}. Proof. move=> a b Da Db; apply: val_inj. by rewrite /= !val_insubd !inE !group1 !groupM ?Da ?Db //= mulg1 gact1. Qed. Canonical sdpair1_morphism := Morphism sdpair1_morphM. Canonical sdpair2_morphism := Morphism sdpair2_morphM. Lemma injm_sdpair1 : 'injm sdpair1. Proof. apply/subsetP=> x /setIP[Rx]. by rewrite !inE -val_eqE val_insubd inE Rx group1 /=; case/andP. Qed. Lemma injm_sdpair2 : 'injm sdpair2. Proof. apply/subsetP=> a /setIP[Da]. by rewrite !inE -val_eqE val_insubd inE Da group1 /=; case/andP. Qed. Lemma sdpairE (u : sdT) : u = sdpair2 u.1 * sdpair1 u.2. Proof. apply: val_inj; case: u => [[a x] /= /setXP[Da Rx]]. by rewrite !val_insubd !inE Da Rx !(group1, gact1) // mulg1 mul1g. Qed. Lemma sdpair_act : {in R & D, forall x a, sdpair1 (to x a) = sdpair1 x ^ sdpair2 a}. Proof. move=> x a Rx Da; apply: val_inj. rewrite /= !val_insubd !inE !group1 gact_stable ?Da ?Rx //=. by rewrite !mul1g mulVg invg1 mulg1 actKVin ?mul1g. Qed. Lemma sdpair_setact (G : {set rT}) a : G \subset R -> a \in D -> sdpair1 @* (to^~ a @: G) = (sdpair1 @* G) :^ sdpair2 a. Proof. move=> sGR Da; have GtoR := subsetP sGR; apply/eqP. rewrite eqEcard cardJg !(card_injm injm_sdpair1) //; last first. by apply/subsetP=> _ /imsetP[x Gx ->]; rewrite gact_stable ?GtoR. rewrite (card_imset _ (act_inj _ _)) leqnn andbT. apply/subsetP=> _ /morphimP[xa Rxa /imsetP[x Gx def_xa ->]]. rewrite mem_conjg -morphV // -sdpair_act ?groupV // def_xa actKin //. by rewrite mem_morphim ?GtoR. Qed. Lemma im_sdpair_norm : sdpair2 @* D \subset 'N(sdpair1 @* R). Proof. apply/subsetP=> _ /morphimP[a _ Da ->]. rewrite inE -sdpair_setact // morphimS //. by apply/subsetP=> _ /imsetP[x Rx ->]; rewrite gact_stable. Qed. Lemma im_sdpair_TI : (sdpair1 @* R) :&: (sdpair2 @* D) = 1. Proof. apply/trivgP; apply/subsetP=> _ /setIP[/morphimP[x _ Rx ->]]. case/morphimP=> a _ Da /eqP; rewrite inE -!val_eqE. by rewrite !val_insubd !inE Da Rx !group1 /eq_op /= eqxx; case/andP. Qed. Lemma im_sdpair : (sdpair1 @* R) * (sdpair2 @* D) = setT. Proof. apply/eqP; rewrite -subTset -(normC im_sdpair_norm). apply/subsetP=> /= u _; rewrite [u]sdpairE. by case: u => [[a x] /= /setXP[Da Rx]]; rewrite mem_mulg ?mem_morphim. Qed. Lemma sdprod_sdpair : sdpair1 @* R ><| sdpair2 @* D = setT. Proof. by rewrite sdprodE ?(im_sdpair_norm, im_sdpair, im_sdpair_TI). Qed. Variables (A : {set aT}) (G : {set rT}). Lemma gacentEsd : 'C_(|to)(A) = sdpair1 @*^-1 'C(sdpair2 @* A). Proof. apply/setP=> x; apply/idP/idP. case/setIP=> Rx /afixP cDAx; rewrite mem_morphpre //. apply/centP=> _ /morphimP[a Da Aa ->]; red. by rewrite conjgC -sdpair_act // cDAx // inE Da. case/morphpreP=> Rx cAx; rewrite inE Rx; apply/afixP=> a /setIP[Da Aa]. apply: (injmP injm_sdpair1); rewrite ?gact_stable /= ?sdpair_act //=. by rewrite /conjg (centP cAx) ?mulKg ?mem_morphim. Qed. Hypotheses (sAD : A \subset D) (sGR : G \subset R). Lemma astabEsd : 'C(G | to) = sdpair2 @*^-1 'C(sdpair1 @* G). Proof. have ssGR := subsetP sGR; apply/setP=> a; apply/idP/idP=> [cGa|]. rewrite mem_morphpre ?(astab_dom cGa) //. apply/centP=> _ /morphimP[x Rx Gx ->]; symmetry. by rewrite conjgC -sdpair_act ?(astab_act cGa) ?(astab_dom cGa). case/morphpreP=> Da cGa; rewrite !inE Da; apply/subsetP=> x Gx; rewrite inE. apply/eqP; apply: (injmP injm_sdpair1); rewrite ?gact_stable ?ssGR //=. by rewrite sdpair_act ?ssGR // /conjg -(centP cGa) ?mulKg ?mem_morphim ?ssGR. Qed. Lemma astabsEsd : 'N(G | to) = sdpair2 @*^-1 'N(sdpair1 @* G). Proof. apply/setP=> a; apply/idP/idP=> [nGa|]. have Da := astabs_dom nGa; rewrite mem_morphpre // inE sub_conjg. apply/subsetP=> _ /morphimP[x Rx Gx ->]. by rewrite mem_conjgV -sdpair_act // mem_morphim ?gact_stable ?astabs_act. case/morphpreP=> Da nGa; rewrite !inE Da; apply/subsetP=> x Gx. have Rx := subsetP sGR _ Gx; have Rxa: to x a \in R by rewrite gact_stable. rewrite inE -sub1set -(injmSK injm_sdpair1) ?morphim_set1 ?sub1set //=. by rewrite sdpair_act ?memJ_norm ?mem_morphim. Qed. Lemma actsEsd : [acts A, on G | to] = (sdpair2 @* A \subset 'N(sdpair1 @* G)). Proof. by rewrite sub_morphim_pre -?astabsEsd. Qed. End ExternalSDirProd. Section ProdMorph. Variables gT rT : finGroupType. Implicit Types A B : {set gT}. Implicit Types G H K : {group gT}. Implicit Types C D : {set rT}. Implicit Type L : {group rT}. Section defs. Variables (A B : {set gT}) (fA fB : gT -> FinGroup.sort rT). Definition pprodm of B \subset 'N(A) & {in A & B, morph_act 'J 'J fA fB} & {in A :&: B, fA =1 fB} := fun x => fA (divgr A B x) * fB (remgr A B x). End defs. Section Props. Variables H K : {group gT}. Variables (fH : {morphism H >-> rT}) (fK : {morphism K >-> rT}). Hypothesis nHK : K \subset 'N(H). Hypothesis actf : {in H & K, morph_act 'J 'J fH fK}. Hypothesis eqfHK : {in H :&: K, fH =1 fK}. Local Notation f := (pprodm nHK actf eqfHK). Lemma pprodmE x a : x \in H -> a \in K -> f (x * a) = fH x * fK a. Proof. move=> Hx Ka; have: x * a \in H * K by rewrite mem_mulg. rewrite -remgrP inE /f rcoset_sym mem_rcoset /divgr -mulgA groupMl //. case/andP; move: (remgr H K _) => b Hab Kb; rewrite morphM // -mulgA. have Kab: a * b^-1 \in K by rewrite groupM ?groupV. by congr (_ * _); rewrite eqfHK 1?inE ?Hab // -morphM // mulgKV. Qed. Lemma pprodmEl : {in H, f =1 fH}. Proof. by move=> x Hx; rewrite -(mulg1 x) pprodmE // morph1 !mulg1. Qed. Lemma pprodmEr : {in K, f =1 fK}. Proof. by move=> a Ka; rewrite -(mul1g a) pprodmE // morph1 !mul1g. Qed. Lemma pprodmM : {in H <*> K &, {morph f: x y / x * y}}. Proof. move=> xa yb; rewrite norm_joinEr //. move=> /imset2P[x a Ha Ka ->{xa}] /imset2P[y b Hy Kb ->{yb}]. have Hya: y ^ a^-1 \in H by rewrite -mem_conjg (normsP nHK). rewrite mulgA -(mulgA x) (conjgCV a y) (mulgA x) -mulgA !pprodmE 1?groupMl //. by rewrite morphM // actf ?groupV ?morphV // morphM // !mulgA mulgKV invgK. Qed. Canonical pprodm_morphism := Morphism pprodmM. Lemma morphim_pprodm A B : A \subset H -> B \subset K -> f @* (A * B) = fH @* A * fK @* B. Proof. move=> sAH sBK; rewrite [f @* _]morphimEsub /=; last first. by rewrite norm_joinEr // mulgSS. apply/setP=> y; apply/imsetP/idP=> [[_ /mulsgP[x a Ax Ba ->] ->{y}] |]. have Hx := subsetP sAH x Ax; have Ka := subsetP sBK a Ba. by rewrite pprodmE // imset2_f ?mem_morphim. case/mulsgP=> _ _ /morphimP[x Hx Ax ->] /morphimP[a Ka Ba ->] ->{y}. by exists (x * a); rewrite ?mem_mulg ?pprodmE. Qed. Lemma morphim_pprodml A : A \subset H -> f @* A = fH @* A. Proof. by move=> sAH; rewrite -{1}(mulg1 A) morphim_pprodm ?sub1G // morphim1 mulg1. Qed. Lemma morphim_pprodmr B : B \subset K -> f @* B = fK @* B. Proof. by move=> sBK; rewrite -{1}(mul1g B) morphim_pprodm ?sub1G // morphim1 mul1g. Qed. Lemma ker_pprodm : 'ker f = [set x * a^-1 | x in H, a in K & fH x == fK a]. Proof. apply/setP=> y; rewrite 3!inE {1}norm_joinEr //=. apply/andP/imset2P=> [[/mulsgP[x a Hx Ka ->{y}]]|[x a Hx]]. rewrite pprodmE // => fxa1. by exists x a^-1; rewrite ?invgK // inE groupVr ?morphV // eq_mulgV1 invgK. case/setIdP=> Kx /eqP fx ->{y}. by rewrite imset2_f ?pprodmE ?groupV ?morphV // fx mulgV. Qed. Lemma injm_pprodm : 'injm f = [&& 'injm fH, 'injm fK & fH @* H :&: fK @* K == fH @* K]. Proof. apply/idP/and3P=> [injf | [injfH injfK]]. rewrite eq_sym -{1}morphimIdom -(morphim_pprodml (subsetIl _ _)) injmI //. rewrite morphim_pprodml // morphim_pprodmr //=; split=> //. apply/injmP=> x y Hx Hy /=; rewrite -!pprodmEl //. by apply: (injmP injf); rewrite ?mem_gen ?inE ?Hx ?Hy. apply/injmP=> a b Ka Kb /=; rewrite -!pprodmEr //. by apply: (injmP injf); rewrite ?mem_gen //; apply/setUP; right. move/eqP=> fHK; rewrite ker_pprodm; apply/subsetP=> y. case/imset2P=> x a Hx /setIdP[Ka /eqP fxa] ->. have: fH x \in fH @* K by rewrite -fHK inE {2}fxa !mem_morphim. case/morphimP=> z Hz Kz /(injmP injfH) def_x. rewrite def_x // eqfHK ?inE ?Hz // in fxa. by rewrite def_x // (injmP injfK _ _ Kz Ka fxa) mulgV set11. Qed. End Props. Section Sdprodm. Variables H K G : {group gT}. Variables (fH : {morphism H >-> rT}) (fK : {morphism K >-> rT}). Hypothesis eqHK_G : H ><| K = G. Hypothesis actf : {in H & K, morph_act 'J 'J fH fK}. Lemma sdprodm_norm : K \subset 'N(H). Proof. by case/sdprodP: eqHK_G. Qed. Lemma sdprodm_sub : G \subset H <*> K. Proof. by case/sdprodP: eqHK_G => _ <- nHK _; rewrite norm_joinEr. Qed. Lemma sdprodm_eqf : {in H :&: K, fH =1 fK}. Proof. by case/sdprodP: eqHK_G => _ _ _ -> _ /set1P->; rewrite !morph1. Qed. Definition sdprodm := restrm sdprodm_sub (pprodm sdprodm_norm actf sdprodm_eqf). Canonical sdprodm_morphism := Eval hnf in [morphism of sdprodm]. Lemma sdprodmE a b : a \in H -> b \in K -> sdprodm (a * b) = fH a * fK b. Proof. exact: pprodmE. Qed. Lemma sdprodmEl a : a \in H -> sdprodm a = fH a. Proof. exact: pprodmEl. Qed. Lemma sdprodmEr b : b \in K -> sdprodm b = fK b. Proof. exact: pprodmEr. Qed. Lemma morphim_sdprodm A B : A \subset H -> B \subset K -> sdprodm @* (A * B) = fH @* A * fK @* B. Proof. move=> sAH sBK; rewrite morphim_restrm /= (setIidPr _) ?morphim_pprodm //. by case/sdprodP: eqHK_G => _ <- _ _; apply: mulgSS. Qed. Lemma im_sdprodm : sdprodm @* G = fH @* H * fK @* K. Proof. by rewrite -morphim_sdprodm //; case/sdprodP: eqHK_G => _ ->. Qed. Lemma morphim_sdprodml A : A \subset H -> sdprodm @* A = fH @* A. Proof. by move=> sHA; rewrite -{1}(mulg1 A) morphim_sdprodm ?sub1G // morphim1 mulg1. Qed. Lemma morphim_sdprodmr B : B \subset K -> sdprodm @* B = fK @* B. Proof. by move=> sBK; rewrite -{1}(mul1g B) morphim_sdprodm ?sub1G // morphim1 mul1g. Qed. Lemma ker_sdprodm : 'ker sdprodm = [set a * b^-1 | a in H, b in K & fH a == fK b]. Proof. rewrite ker_restrm (setIidPr _) ?subIset ?ker_pprodm //; apply/orP; left. by case/sdprodP: eqHK_G => _ <- nHK _; rewrite norm_joinEr. Qed. Lemma injm_sdprodm : 'injm sdprodm = [&& 'injm fH, 'injm fK & fH @* H :&: fK @* K == 1]. Proof. rewrite ker_sdprodm -(ker_pprodm sdprodm_norm actf sdprodm_eqf) injm_pprodm. congr [&& _, _ & _ == _]; have [_ _ _ tiHK] := sdprodP eqHK_G. by rewrite -morphimIdom tiHK morphim1. Qed. End Sdprodm. Section Cprodm. Variables H K G : {group gT}. Variables (fH : {morphism H >-> rT}) (fK : {morphism K >-> rT}). Hypothesis eqHK_G : H \* K = G. Hypothesis cfHK : fK @* K \subset 'C(fH @* H). Hypothesis eqfHK : {in H :&: K, fH =1 fK}. Lemma cprodm_norm : K \subset 'N(H). Proof. by rewrite cents_norm //; case/cprodP: eqHK_G. Qed. Lemma cprodm_sub : G \subset H <*> K. Proof. by case/cprodP: eqHK_G => _ <- cHK; rewrite cent_joinEr. Qed. Lemma cprodm_actf : {in H & K, morph_act 'J 'J fH fK}. Proof. case/cprodP: eqHK_G => _ _ cHK a b Ha Kb /=. by rewrite /conjg -(centsP cHK b) // -(centsP cfHK (fK b)) ?mulKg ?mem_morphim. Qed. Definition cprodm := restrm cprodm_sub (pprodm cprodm_norm cprodm_actf eqfHK). Canonical cprodm_morphism := Eval hnf in [morphism of cprodm]. Lemma cprodmE a b : a \in H -> b \in K -> cprodm (a * b) = fH a * fK b. Proof. exact: pprodmE. Qed. Lemma cprodmEl a : a \in H -> cprodm a = fH a. Proof. exact: pprodmEl. Qed. Lemma cprodmEr b : b \in K -> cprodm b = fK b. Proof. exact: pprodmEr. Qed. Lemma morphim_cprodm A B : A \subset H -> B \subset K -> cprodm @* (A * B) = fH @* A * fK @* B. Proof. move=> sAH sBK; rewrite morphim_restrm /= (setIidPr _) ?morphim_pprodm //. by case/cprodP: eqHK_G => _ <- _; apply: mulgSS. Qed. Lemma im_cprodm : cprodm @* G = fH @* H * fK @* K. Proof. by have [_ defHK _] := cprodP eqHK_G; rewrite -{2}defHK morphim_cprodm. Qed. Lemma morphim_cprodml A : A \subset H -> cprodm @* A = fH @* A. Proof. by move=> sHA; rewrite -{1}(mulg1 A) morphim_cprodm ?sub1G // morphim1 mulg1. Qed. Lemma morphim_cprodmr B : B \subset K -> cprodm @* B = fK @* B. Proof. by move=> sBK; rewrite -{1}(mul1g B) morphim_cprodm ?sub1G // morphim1 mul1g. Qed. Lemma ker_cprodm : 'ker cprodm = [set a * b^-1 | a in H, b in K & fH a == fK b]. Proof. rewrite ker_restrm (setIidPr _) ?subIset ?ker_pprodm //; apply/orP; left. by case/cprodP: eqHK_G => _ <- cHK; rewrite cent_joinEr. Qed. Lemma injm_cprodm : 'injm cprodm = [&& 'injm fH, 'injm fK & fH @* H :&: fK @* K == fH @* K]. Proof. by rewrite ker_cprodm -(ker_pprodm cprodm_norm cprodm_actf eqfHK) injm_pprodm. Qed. End Cprodm. Section Dprodm. Variables G H K : {group gT}. Variables (fH : {morphism H >-> rT}) (fK : {morphism K >-> rT}). Hypothesis eqHK_G : H \x K = G. Hypothesis cfHK : fK @* K \subset 'C(fH @* H). Lemma dprodm_cprod : H \* K = G. Proof. by rewrite -eqHK_G /dprod; case/dprodP: eqHK_G => _ _ _ ->; rewrite subxx. Qed. Lemma dprodm_eqf : {in H :&: K, fH =1 fK}. Proof. by case/dprodP: eqHK_G => _ _ _ -> _ /set1P->; rewrite !morph1. Qed. Definition dprodm := cprodm dprodm_cprod cfHK dprodm_eqf. Canonical dprodm_morphism := Eval hnf in [morphism of dprodm]. Lemma dprodmE a b : a \in H -> b \in K -> dprodm (a * b) = fH a * fK b. Proof. exact: pprodmE. Qed. Lemma dprodmEl a : a \in H -> dprodm a = fH a. Proof. exact: pprodmEl. Qed. Lemma dprodmEr b : b \in K -> dprodm b = fK b. Proof. exact: pprodmEr. Qed. Lemma morphim_dprodm A B : A \subset H -> B \subset K -> dprodm @* (A * B) = fH @* A * fK @* B. Proof. exact: morphim_cprodm. Qed. Lemma im_dprodm : dprodm @* G = fH @* H * fK @* K. Proof. exact: im_cprodm. Qed. Lemma morphim_dprodml A : A \subset H -> dprodm @* A = fH @* A. Proof. exact: morphim_cprodml. Qed. Lemma morphim_dprodmr B : B \subset K -> dprodm @* B = fK @* B. Proof. exact: morphim_cprodmr. Qed. Lemma ker_dprodm : 'ker dprodm = [set a * b^-1 | a in H, b in K & fH a == fK b]. Proof. exact: ker_cprodm. Qed. Lemma injm_dprodm : 'injm dprodm = [&& 'injm fH, 'injm fK & fH @* H :&: fK @* K == 1]. Proof. rewrite injm_cprodm -(morphimIdom fH K). by case/dprodP: eqHK_G => _ _ _ ->; rewrite morphim1. Qed. End Dprodm. Lemma isog_dprod A B G C D L : A \x B = G -> C \x D = L -> isog A C -> isog B D -> isog G L. Proof. move=> defG {C D} /dprodP[[C D -> ->] defL cCD trCD]. case/dprodP: defG (defG) => {A B} [[A B -> ->] defG _ _] dG defC defD. case/isogP: defC defL cCD trCD => fA injfA <-{C}. case/isogP: defD => fB injfB <-{D} defL cCD trCD. apply/isogP; exists (dprodm_morphism dG cCD). by rewrite injm_dprodm injfA injfB trCD eqxx. by rewrite /= -{2}defG morphim_dprodm. Qed. End ProdMorph. Section ExtSdprodm. Variables gT aT rT : finGroupType. Variables (H : {group gT}) (K : {group aT}) (to : groupAction K H). Variables (fH : {morphism H >-> rT}) (fK : {morphism K >-> rT}). Hypothesis actf : {in H & K, morph_act to 'J fH fK}. Local Notation fsH := (fH \o invm (injm_sdpair1 to)). Local Notation fsK := (fK \o invm (injm_sdpair2 to)). Let DgH := sdpair1 to @* H. Let DgK := sdpair2 to @* K. Lemma xsdprodm_dom1 : DgH \subset 'dom fsH. Proof. by rewrite ['dom _]morphpre_invm. Qed. Local Notation gH := (restrm xsdprodm_dom1 fsH). Lemma xsdprodm_dom2 : DgK \subset 'dom fsK. Proof. by rewrite ['dom _]morphpre_invm. Qed. Local Notation gK := (restrm xsdprodm_dom2 fsK). Lemma im_sdprodm1 : gH @* DgH = fH @* H. Proof. by rewrite morphim_restrm setIid morphim_comp im_invm. Qed. Lemma im_sdprodm2 : gK @* DgK = fK @* K. Proof. by rewrite morphim_restrm setIid morphim_comp im_invm. Qed. Lemma xsdprodm_act : {in DgH & DgK, morph_act 'J 'J gH gK}. Proof. move=> fh fk; case/morphimP=> h _ Hh ->{fh}; case/morphimP=> k _ Kk ->{fk}. by rewrite /= -sdpair_act // /restrm /= !invmE ?actf ?gact_stable. Qed. Definition xsdprodm := sdprodm (sdprod_sdpair to) xsdprodm_act. Canonical xsdprod_morphism := [morphism of xsdprodm]. Lemma im_xsdprodm : xsdprodm @* setT = fH @* H * fK @* K. Proof. by rewrite -im_sdpair morphim_sdprodm // im_sdprodm1 im_sdprodm2. Qed. Lemma injm_xsdprodm : 'injm xsdprodm = [&& 'injm fH, 'injm fK & fH @* H :&: fK @* K == 1]. Proof. rewrite injm_sdprodm im_sdprodm1 im_sdprodm2 !subG1 /= !ker_restrm !ker_comp. rewrite !morphpre_invm !morphimIim. by rewrite !morphim_injm_eq1 ?subsetIl ?injm_sdpair1 ?injm_sdpair2. Qed. End ExtSdprodm. Section DirprodIsom. Variable gT : finGroupType. Implicit Types G H : {group gT}. Definition mulgm : gT * gT -> _ := uncurry mulg. Lemma imset_mulgm (A B : {set gT}) : mulgm @: setX A B = A * B. Proof. by rewrite -curry_imset2X. Qed. Lemma mulgmP H1 H2 G : reflect (H1 \x H2 = G) (misom (setX H1 H2) G mulgm). Proof. apply: (iffP misomP) => [[pM /isomP[injf /= <-]] | ]. have /dprodP[_ /= defX cH12] := setX_dprod H1 H2. rewrite -{4}defX {}defX => /(congr1 (fun A => morphm pM @* A)). move/(morphimS (morphm_morphism pM)): cH12 => /=. have sH1H: setX H1 1 \subset setX H1 H2 by rewrite setXS ?sub1G. have sH2H: setX 1 H2 \subset setX H1 H2 by rewrite setXS ?sub1G. rewrite morphim1 injm_cent ?injmI //= subsetI => /andP[_]. by rewrite !morphimEsub //= !imset_mulgm mulg1 mul1g; apply: dprodE. case/dprodP=> _ defG cH12 trH12. have fM: morphic (setX H1 H2) mulgm. apply/morphicP=> [[x1 x2] [y1 y2] /setXP[_ Hx2] /setXP[Hy1 _]]. by rewrite /= mulgA -(mulgA x1) -(centsP cH12 x2) ?mulgA. exists fM; apply/isomP; split; last by rewrite morphimEsub //= imset_mulgm. apply/subsetP=> [[x1 x2]]; rewrite !inE /= andbC -eq_invg_mul. case: eqP => //= <-; rewrite groupV -in_setI trH12 => /set1P->. by rewrite invg1 eqxx. Qed. End DirprodIsom. Arguments mulgmP {gT H1 H2 G}. Prenex Implicits mulgm.