fact stringlengths 8 1.54k | type stringclasses 19 values | library stringclasses 8 values | imports listlengths 1 10 | filename stringclasses 98 values | symbolic_name stringlengths 1 42 | docstring stringclasses 1 value |
|---|---|---|---|---|---|---|
isog_Mho(H : {group rT}) : G \isog H -> 'Mho^n(G) \isog 'Mho^n(H).
Proof. exact: gFisog. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | isog_Mho | |
Ohm0G : 'Ohm_0(G) = 1.
Proof.
by apply/trivgP; rewrite /= gen_subG; apply/subsetP=> x /setIdP[_] /[1!inE].
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | Ohm0 | |
Ohm_leqm n G : m <= n -> 'Ohm_m(G) \subset 'Ohm_n(G).
Proof.
move/subnKC <-; rewrite genS //; apply/subsetP=> y.
by rewrite !inE expnD expgM => /andP[-> /eqP->]; rewrite expg1n /=.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | Ohm_leq | |
OhmJn G x : 'Ohm_n(G :^ x) = 'Ohm_n(G) :^ x.
Proof.
rewrite -{1}(setIid G) -(setIidPr (Ohm_sub n G)).
by rewrite -!morphim_conj injm_Ohm ?injm_conj.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | OhmJ | |
Mho0G : 'Mho^0(G) = G.
Proof.
apply/eqP; rewrite eqEsubset Mho_sub /=.
apply/subsetP=> x Gx; rewrite -[x]prod_constt group_prod // => p _.
exact: Mho_p_elt (groupX _ Gx) (p_elt_constt _ _).
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | Mho0 | |
Mho_leqm n G : m <= n -> 'Mho^n(G) \subset 'Mho^m(G).
Proof.
move/subnKC <-; rewrite gen_subG //.
apply/subsetP=> _ /imsetP[x /setIdP[Gx p_x] ->].
by rewrite expnD expgM groupX ?(Mho_p_elt _ _ p_x).
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | Mho_leq | |
MhoJn G x : 'Mho^n(G :^ x) = 'Mho^n(G) :^ x.
Proof.
by rewrite -{1}(setIid G) -(setIidPr (Mho_sub n G)) -!morphim_conj morphim_Mho.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | MhoJ | |
extend_cyclic_MhoG p x :
p.-group G -> x \in G -> 'Mho^1(G) = <[x ^+ p]> ->
forall k, k > 0 -> 'Mho^k(G) = <[x ^+ (p ^ k)]>.
Proof.
move=> pG Gx defG1 [//|k _]; have pX := mem_p_elt pG Gx.
apply/eqP; rewrite eqEsubset cycle_subG (Mho_p_elt _ Gx pX) andbT.
rewrite (MhoE _ pG) gen_subG; apply/subsetP=> ypk; case/imsetP=> y Gy ->{ypk}.
have: y ^+ p \in <[x ^+ p]> by rewrite -defG1 (Mho_p_elt 1 _ (mem_p_elt pG Gy)).
rewrite !expnS /= !expgM => /cycleP[j ->].
by rewrite -!expgM mulnCA mulnC expgM mem_cycle.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | extend_cyclic_Mho | |
Ohm1EprimeG : 'Ohm_1(G) = <<[set x in G | prime #[x]]>>.
Proof.
rewrite -['Ohm_1(G)](genD1 (group1 _)); congr <<_>>.
apply/setP=> x; rewrite !inE andbCA -order_dvdn -order_gt1; congr (_ && _).
apply/andP/idP=> [[p_gt1] | p_pr]; last by rewrite prime_gt1 ?pdiv_id.
set p := pdiv _ => ox_p; have p_pr: prime p by rewrite pdiv_prime.
by have [_ dv_p] := primeP p_pr; case/pred2P: (dv_p _ ox_p) p_gt1 => ->.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | Ohm1Eprime | |
abelem_Ohm1p G : p.-group G -> p.-abelem 'Ohm_1(G) = abelian 'Ohm_1(G).
Proof.
move=> pG; rewrite /abelem (pgroupS (Ohm_sub 1 G)) //.
case abG1: (abelian _) => //=; apply/exponentP=> x.
by rewrite (OhmEabelian pG abG1); case/LdivP.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelem_Ohm1 | |
Ohm1_abelemp G : p.-group G -> abelian G -> p.-abelem ('Ohm_1(G)).
Proof. by move=> pG cGG; rewrite abelem_Ohm1 ?(abelianS (Ohm_sub 1 G)). Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | Ohm1_abelem | |
Ohm1_idp G : p.-abelem G -> 'Ohm_1(G) = G.
Proof.
case/and3P=> pG cGG /exponentP Gp.
apply/eqP; rewrite eqEsubset Ohm_sub (OhmE 1 pG) sub_gen //.
by apply/subsetP=> x Gx; rewrite !inE Gx Gp /=.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | Ohm1_id | |
abelem_Ohm1Pp G :
abelian G -> p.-group G -> reflect ('Ohm_1(G) = G) (p.-abelem G).
Proof.
move=> cGG pG.
by apply: (iffP idP) => [| <-]; [apply: Ohm1_id | apply: Ohm1_abelem].
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelem_Ohm1P | |
TI_Ohm1G H : H :&: 'Ohm_1(G) = 1 -> H :&: G = 1.
Proof.
move=> tiHG1; case: (trivgVpdiv (H :&: G)) => // [[p pr_p]].
case/Cauchy=> // x /setIP[Hx Gx] ox.
suffices x1: x \in [1] by rewrite -ox (set1P x1) order1 in pr_p.
by rewrite -{}tiHG1 inE Hx Ohm1Eprime mem_gen // inE Gx ox.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | TI_Ohm1 | |
Ohm1_eq1G : ('Ohm_1(G) == 1) = (G :==: 1).
Proof.
apply/idP/idP => [/eqP G1_1 | /eqP->]; last by rewrite -subG1 Ohm_sub.
by rewrite -(setIid G) TI_Ohm1 // G1_1 setIg1.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | Ohm1_eq1 | |
meet_Ohm1G H : G :&: H != 1 -> G :&: 'Ohm_1(H) != 1.
Proof. by apply: contraNneq => /TI_Ohm1->. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | meet_Ohm1 | |
Ohm1_cent_maxG E p : E \in 'E*_p(G) -> p.-group G -> 'Ohm_1('C_G(E)) = E.
Proof.
move=> EpmE pG; have [G1 | ntG]:= eqsVneq G 1.
case/pmaxElemP: EpmE; case/pElemP; rewrite G1 => /trivgP-> _ _.
by apply/trivgP; rewrite cent1T setIT Ohm_sub.
have [p_pr _ _] := pgroup_pdiv pG ntG.
by rewrite (OhmE 1 (pgroupS (subsetIl G _) pG)) (pmaxElem_LdivP _ _) ?genGid.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | Ohm1_cent_max | |
Ohm1_cyclic_pgroup_primep G :
cyclic G -> p.-group G -> G :!=: 1 -> #|'Ohm_1(G)| = p.
Proof.
move=> cycG pG ntG; set K := 'Ohm_1(G).
have abelK: p.-abelem K by rewrite Ohm1_abelem ?cyclic_abelian.
have sKG: K \subset G := Ohm_sub 1 G.
case/cyclicP: (cyclicS sKG cycG) => x /=; rewrite -/K => defK.
rewrite defK -orderE (abelem_order_p abelK) //= -/K ?defK ?cycle_id //.
rewrite -cycle_eq1 -defK -(setIidPr sKG).
by apply: contraNneq ntG => /TI_Ohm1; rewrite setIid => ->.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | Ohm1_cyclic_pgroup_prime | |
cyclic_pgroup_dprod_trivgp A B C :
p.-group C -> cyclic C -> A \x B = C ->
A = 1 /\ B = C \/ B = 1 /\ A = C.
Proof.
move=> pC cycC; case/cyclicP: cycC pC => x ->{C} pC defC.
case/dprodP: defC => [] [G H -> ->{A B}] defC _ tiGH; rewrite -defC.
have [/trivgP | ntC] := eqVneq <[x]> 1.
by rewrite -defC mulG_subG => /andP[/trivgP-> _]; rewrite mul1g; left.
have [pr_p _ _] := pgroup_pdiv pC ntC; pose K := 'Ohm_1(<[x]>).
have prK : prime #|K| by rewrite (Ohm1_cyclic_pgroup_prime _ pC) ?cycle_cyclic.
case: (prime_subgroupVti G prK) => [sKG |]; last first.
move/TI_Ohm1; rewrite -defC (setIidPl (mulG_subl _ _)) => ->.
by left; rewrite mul1g.
case: (prime_subgroupVti H prK) => [sKH |]; last first.
move/TI_Ohm1; rewrite -defC (setIidPl (mulG_subr _ _)) => ->.
by right; rewrite mulg1.
have K1: K :=: 1 by apply/trivgP; rewrite -tiGH subsetI sKG.
by rewrite K1 cards1 in prK.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | cyclic_pgroup_dprod_trivg | |
piOhm1G : \pi('Ohm_1(G)) = \pi(G).
Proof.
apply/eq_piP => p; apply/idP/idP; first exact: (piSg (Ohm_sub 1 G)).
rewrite !mem_primes !cardG_gt0 => /andP[p_pr /Cauchy[] // x Gx oxp].
by rewrite p_pr -oxp order_dvdG //= Ohm1Eprime mem_gen // inE Gx oxp.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | piOhm1 | |
Ohm1Eexponentp G :
prime p -> exponent 'Ohm_1(G) %| p -> 'Ohm_1(G) = 'Ldiv_p(G).
Proof.
move=> p_pr expG1p; have pG: p.-group G.
apply: sub_in_pnat (pnat_pi (cardG_gt0 G)) => q _.
rewrite -piOhm1 mem_primes; case/and3P=> q_pr _; apply: pgroupP q_pr.
by rewrite -pnat_exponent (pnat_dvd expG1p) ?pnat_id.
apply/eqP; rewrite eqEsubset {2}(OhmE 1 pG) subset_gen subsetI Ohm_sub.
by rewrite sub_LdivT expG1p.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | Ohm1Eexponent | |
p_rank_Ohm1p G : 'r_p('Ohm_1(G)) = 'r_p(G).
Proof.
apply/eqP; rewrite eqn_leq p_rankS ?Ohm_sub //.
apply/bigmax_leqP=> E /setIdP[sEG abelE].
by rewrite (bigmax_sup E) // inE -{1}(Ohm1_id abelE) OhmS.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | p_rank_Ohm1 | |
rank_Ohm1G : 'r('Ohm_1(G)) = 'r(G).
Proof.
apply/eqP; rewrite eqn_leq rankS ?Ohm_sub //.
by have [p _ ->] := rank_witness G; rewrite -p_rank_Ohm1 p_rank_le_rank.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | rank_Ohm1 | |
p_rank_abelianp G : abelian G -> 'r_p(G) = logn p #|'Ohm_1(G)|.
Proof.
move=> cGG; have nilG := abelian_nil cGG; case p_pr: (prime p); last first.
by apply/eqP; rewrite lognE p_pr eqn0Ngt p_rank_gt0 mem_primes p_pr.
case/dprodP: (Ohm_dprod 1 (nilpotent_pcoreC p nilG)) => _ <- _ /TI_cardMg->.
rewrite mulnC logn_Gauss; last first.
rewrite prime_coprime // -p'natE // -/(pgroup _ _).
exact: pgroupS (Ohm_sub _ _) (pcore_pgroup _ _).
rewrite -(p_rank_Sylow (nilpotent_pcore_Hall p nilG)) -p_rank_Ohm1.
rewrite p_rank_abelem // Ohm1_abelem ?pcore_pgroup //.
exact: abelianS (pcore_sub _ _) cGG.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | p_rank_abelian | |
rank_abelian_pgroupp G :
p.-group G -> abelian G -> 'r(G) = logn p #|'Ohm_1(G)|.
Proof. by move=> pG cGG; rewrite (rank_pgroup pG) p_rank_abelian. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | rank_abelian_pgroup | |
abelian_splitsx G :
x \in G -> #[x] = exponent G -> abelian G -> [splits G, over <[x]>].
Proof.
move=> Gx ox cGG; apply/splitsP; have [n] := ubnP #|G|.
elim: n gT => // n IHn aT in x G Gx ox cGG * => /ltnSE-leGn.
have: <[x]> \subset G by [rewrite cycle_subG]; rewrite subEproper.
case/predU1P=> [<- | /properP[sxG [y]]].
by exists 1%G; rewrite inE -subG1 subsetIr mulg1 /=.
have [m] := ubnP #[y]; elim: m y => // m IHm y /ltnSE-leym Gy x'y.
case: (trivgVpdiv <[y]>) => [y1 | [p p_pr p_dv_y]].
by rewrite -cycle_subG y1 sub1G in x'y.
case x_yp: (y ^+ p \in <[x]>); last first.
apply: IHm (negbT x_yp); rewrite ?groupX ?(leq_trans _ leym) //.
by rewrite orderXdiv // ltn_Pdiv ?prime_gt1.
have{x_yp} xp_yp: (y ^+ p \in <[x ^+ p]>).
have: <[y ^+ p]>%G \in [set <[x ^+ (#[x] %/ #[y ^+ p])]>%G].
by rewrite -cycle_sub_group ?order_dvdG // inE cycle_subG x_yp eqxx.
rewrite inE -cycle_subG -val_eqE /=; move/eqP->.
rewrite cycle_subG orderXdiv // divnA // mulnC ox.
by rewrite -muln_divA ?dvdn_exponent ?expgM 1?groupX ?cycle_id.
have: p <= #[y] by rewrite dvdn_leq.
rewrite leq_eqVlt => /predU1P[{xp_yp m IHm leym}oy | ltpy]; last first.
case/cycleP: xp_yp => k; rewrite -expgM mulnC expgM => def_yp.
suffices: #[y * x ^- k] < m.
by move/IHm; apply; rewrite groupMr // groupV groupX ?cycle_id.
apply: leq_ltn_trans (leq_trans ltpy leym).
rewrite dvdn_leq ?prime_gt0 // order_dvdn expgMn.
by rewrite expgVn def_yp mulgV.
by apply: (centsP cGG); rewrite ?groupV ?groupX.
pose Y := <
... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelian_splits | |
abelem_splitsp G H : p.-abelem G -> H \subset G -> [splits G, over H].
Proof.
have [m] := ubnP #|G|; elim: m G H => // m IHm G H /ltnSE-leGm abelG sHG.
have [-> | ] := eqsVneq H 1.
by apply/splitsP; exists G; rewrite inE mul1g -subG1 subsetIl /=.
case/trivgPn=> x Hx ntx; have Gx := subsetP sHG x Hx.
have [_ cGG eGp] := and3P abelG.
have ox: #[x] = exponent G.
by apply/eqP; rewrite eqn_dvd dvdn_exponent // (abelem_order_p abelG).
case/splitsP: (abelian_splits Gx ox cGG) => K; case/complP=> tixK defG.
have sKG: K \subset G by rewrite -defG mulG_subr.
have ltKm: #|K| < m.
rewrite (leq_trans _ leGm) ?proper_card //; apply/properP; split=> //.
exists x => //; apply: contra ntx => Kx; rewrite -cycle_eq1 -subG1 -tixK.
by rewrite subsetI subxx cycle_subG.
case/splitsP: (IHm _ _ ltKm (abelemS sKG abelG) (subsetIr H K)) => L.
case/complP=> tiHKL defK; apply/splitsP; exists L; rewrite inE.
rewrite -subG1 -tiHKL -setIA setIS; last by rewrite subsetI -defK mulG_subr /=.
by rewrite -(setIidPr sHG) -defG -group_modl ?cycle_subG //= setIC -mulgA defK.
Qed.
Fact abelian_type_subproof G :
{H : {group gT} & abelian G -> {x | #[x] = exponent G & <[x]> \x H = G}}.
Proof.
case cGG: (abelian G); last by exists G.
have [x Gx ox] := exponent_witness (abelian_nil cGG).
case/splitsP/ex_mingroup: (abelian_splits Gx (esym ox) cGG) => H.
case/mingroupp/complP=> tixH defG; exists H => _.
exists x; rewrite ?dprodE // (sub_abelian_cent2 cGG) ?cycle_subG //.
by rewrite -defG mulG_subr.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelem_splits | |
abelian_type_recn G :=
if n is n'.+1 then if abelian G && (G :!=: 1) then
exponent G :: abelian_type_rec n' (tag (abelian_type_subproof G))
else [::] else [::]. | Fixpoint | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelian_type_rec | |
abelian_type(A : {set gT}) := abelian_type_rec #|A| <<A>>. | Definition | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelian_type | |
abelian_type_dvdn_sortedA : sorted [rel m n | n %| m] (abelian_type A).
Proof.
set R := SimplRel _; pose G := <<A>>%G; pose M := G.
suffices: path R (exponent M) (abelian_type A) by case: (_ A) => // m t /andP[].
rewrite /abelian_type -/G; have: G \subset M by [].
elim: {A}#|A| G M => //= n IHn G M sGM.
case: andP => //= -[cGG ntG]; rewrite exponentS ?IHn //=.
case: (abelian_type_subproof G) => H /= [//| x _] /dprodP[_ /= <- _ _].
exact: mulG_subr.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelian_type_dvdn_sorted | |
abelian_type_gt1A : all [pred m | m > 1] (abelian_type A).
Proof.
rewrite /abelian_type; elim: {A}#|A| <<A>>%G => //= n IHn G.
case: ifP => //= /andP[_ ntG]; rewrite {n}IHn.
by rewrite ltn_neqAle exponent_gt0 eq_sym -dvdn1 -trivg_exponent ntG.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelian_type_gt1 | |
abelian_type_sortedA : sorted geq (abelian_type A).
Proof.
have:= abelian_type_dvdn_sorted A; have:= abelian_type_gt1 A.
case: (abelian_type A) => //= m t; elim: t m => //= n t IHt m /andP[].
by move/ltnW=> m_gt0 t_gt1 /andP[n_dv_m /IHt->]; rewrite // dvdn_leq.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelian_type_sorted | |
abelian_structureG :
abelian G ->
{b | \big[dprod/1]_(x <- b) <[x]> = G & map order b = abelian_type G}.
Proof.
rewrite /abelian_type genGidG; have [n] := ubnPleq #|G|.
elim: n G => /= [|n IHn] G leGn cGG; first by rewrite leqNgt cardG_gt0 in leGn.
rewrite [in _ && _]cGG /=; case: ifP => [ntG|/eqP->]; last first.
by exists [::]; rewrite ?big_nil.
case: (abelian_type_subproof G) => H /= [//|x ox xdefG]; rewrite -ox.
have [_ defG cxH tixH] := dprodP xdefG.
have sHG: H \subset G by rewrite -defG mulG_subr.
case/IHn: (abelianS sHG cGG) => [|b defH <-].
rewrite -ltnS (leq_trans _ leGn) // -defG TI_cardMg // -orderE.
rewrite ltn_Pmull ?cardG_gt0 // ltn_neqAle order_gt0 eq_sym -dvdn1.
by rewrite ox -trivg_exponent ntG.
by exists (x :: b); rewrite // big_cons defH xdefG.
Qed. | Theorem | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelian_structure | |
count_logn_dprod_cyclep n b G :
\big[dprod/1]_(x <- b) <[x]> = G ->
count [pred x | logn p #[x] > n] b = logn p #|'Ohm_n.+1(G) : 'Ohm_n(G)|.
Proof.
have sOn1 H: 'Ohm_n(H) \subset 'Ohm_n.+1(H) by apply: Ohm_leq.
pose lnO i (A : {set gT}) := logn p #|'Ohm_i(A)|.
have lnO_le H: lnO n H <= lnO n.+1 H.
by rewrite dvdn_leq_log ?cardG_gt0 // cardSg ?sOn1.
have lnOx i A B H: A \x B = H -> lnO i A + lnO i B = lnO i H.
move=> defH; case/dprodP: defH (defH) => {A B}[[A B -> ->]] _ _ _ defH.
rewrite /lnO; case/dprodP: (Ohm_dprod i defH) => _ <- _ tiOAB.
by rewrite TI_cardMg ?lognM.
rewrite -divgS //= logn_div ?cardSg //= -/(lnO _ _) -/(lnO _ _).
elim: b G => [_ <-|x b IHb G] /=.
by rewrite big_nil /lnO !(trivgP (Ohm_sub _ _)) subnn.
rewrite /= big_cons => defG; rewrite -!(lnOx _ _ _ _ defG) subnDA.
case/dprodP: defG => [[_ H _ defH] _ _ _] {G}; rewrite defH (IHb _ defH).
symmetry; do 2!rewrite addnC -addnBA ?lnO_le //; congr (_ + _).
pose y := x.`_p; have p_y: p.-elt y by rewrite p_elt_constt.
have{lnOx} lnOy i: lnO i <[x]> = lnO i <[y]>.
have cXX := cycle_abelian x.
have co_yx': coprime #[y] #[x.`_p^'] by rewrite !order_constt coprime_partC.
have defX: <[y]> \x <[x.`_p^']> = <[x]>.
rewrite dprodE ?coprime_TIg //.
by rewrite -cycleM ?consttC //; apply: (centsP cXX); apply: mem_cycle.
by apply: (sub_abelian_cent2 cXX); rewrite cycle_subG mem_cycle.
rewrite -(lnOx i _ _ _ defX) addnC {1}/lnO lognE.
case: and3P => // [[p_pr _ /idPn[]]]; rewrite -p'natE //.
exact: pgroupS
... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | count_logn_dprod_cycle | |
abelian_type_pgroupp b G :
p.-group G -> \big[dprod/1]_(x <- b) <[x]> = G -> 1 \notin b ->
perm_eq (abelian_type G) (map order b).
Proof.
rewrite perm_sym; move: b => b1 pG defG1 ntb1.
have cGG: abelian G.
elim: (b1) {pG}G defG1 => [_ <-|x b IHb G]; first by rewrite big_nil abelian1.
rewrite big_cons; case/dprodP=> [[_ H _ defH]] <-; rewrite defH => cxH _.
by rewrite abelianM cycle_abelian IHb.
have p_bG b: \big[dprod/1]_(x <- b) <[x]> = G -> all (p_elt p) b.
elim: b {defG1 cGG}G pG => //= x b IHb G pG; rewrite big_cons.
case/dprodP=> [[_ H _ defH]]; rewrite defH andbC => defG _ _.
by rewrite -defG pgroupM in pG; case/andP: pG => p_x /IHb->.
have [b2 defG2 def_t] := abelian_structure cGG.
have ntb2: 1 \notin b2.
apply: contraL (abelian_type_gt1 G) => b2_1.
rewrite -def_t -has_predC has_map.
by apply/hasP; exists 1; rewrite //= order1.
rewrite -{}def_t; apply/allP=> m; rewrite -map_cat => /mapP[x b_x def_m].
have{ntb1 ntb2} ntx: x != 1.
by apply: contraL b_x; move/eqP->; rewrite mem_cat negb_or ntb1 ntb2.
have p_x: p.-elt x by apply: allP (x) b_x; rewrite all_cat !p_bG.
rewrite -cycle_eq1 in ntx; have [p_pr _ [k ox]] := pgroup_pdiv p_x ntx.
apply/eqnP; rewrite {m}def_m orderE ox !count_map.
pose cnt_p k := count [pred x : gT | logn p #[x] > k].
have cnt_b b: \big[dprod/1]_(x <- b) <[x]> = G ->
count [pred x | #[x] == p ^ k.+1]%N b = cnt_p k b - cnt_p k.+1 b.
- move/p_bG; elim: b => //= _ b IHb /andP[/p_natP[j ->] /IHb-> {IHb}].
rewrite eqn_leq !leq_exp2l ?prime_gt1
... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelian_type_pgroup | |
size_abelian_typeG : abelian G -> size (abelian_type G) = 'r(G).
Proof.
move=> cGG; have [b defG def_t] := abelian_structure cGG.
apply/eqP; rewrite -def_t size_map eqn_leq andbC; apply/andP; split.
have [p p_pr ->] := rank_witness G; rewrite p_rank_abelian //.
by rewrite -indexg1 -(Ohm0 G) -(count_logn_dprod_cycle _ _ defG) count_size.
case/lastP def_b: b => // [b' x]; pose p := pdiv #[x].
have p_pr: prime p.
have:= abelian_type_gt1 G; rewrite -def_t def_b map_rcons -cats1 all_cat.
by rewrite /= andbT => /andP[_]; apply: pdiv_prime.
suffices: all [pred y | logn p #[y] > 0] b.
rewrite all_count (count_logn_dprod_cycle _ _ defG) -def_b; move/eqP <-.
by rewrite Ohm0 indexg1 -p_rank_abelian ?p_rank_le_rank.
apply/allP=> y; rewrite def_b mem_rcons inE /= => b_y.
rewrite lognE p_pr order_gt0 (dvdn_trans (pdiv_dvd _)) //.
case/predU1P: b_y => [-> // | b'_y].
have:= abelian_type_dvdn_sorted G; rewrite -def_t def_b.
case/splitPr: b'_y => b1 b2; rewrite -cat_rcons rcons_cat map_cat !map_rcons.
rewrite headI /= cat_path -(last_cons 2) -headI last_rcons.
case/andP=> _ /order_path_min min_y.
apply: (allP (min_y _)) => [? ? ? ? dv|]; first exact: (dvdn_trans dv).
by rewrite mem_rcons mem_head.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | size_abelian_type | |
mul_card_Ohm_Mho_abeliann G :
abelian G -> (#|'Ohm_n(G)| * #|'Mho^n(G)|)%N = #|G|.
Proof.
case/abelian_structure => b defG _.
elim: b G defG => [_ <-|x b IHb G].
by rewrite !big_nil (trivgP (Ohm_sub _ _)) (trivgP (Mho_sub _ _)) !cards1.
rewrite big_cons => defG; rewrite -(dprod_card defG).
rewrite -(dprod_card (Ohm_dprod n defG)) -(dprod_card (Mho_dprod n defG)) /=.
rewrite mulnCA -!mulnA mulnCA mulnA; case/dprodP: defG => [[_ H _ defH] _ _ _].
rewrite defH {b G defH IHb}(IHb H defH); congr (_ * _)%N => {H}.
have [m] := ubnP #[x]; elim: m x => // m IHm x /ltnSE-lexm.
case p_x: (p_group <[x]>); last first.
case: (eqVneq x 1) p_x => [-> |]; first by rewrite cycle1 p_group1.
rewrite -order_gt1 /p_group -orderE; set p := pdiv _ => ntx p'x.
have def_x: <[x.`_p]> \x <[x.`_p^']> = <[x]>.
have ?: coprime #[x.`_p] #[x.`_p^'] by rewrite !order_constt coprime_partC.
have ?: commute x.`_p x.`_p^' by apply: commuteX2.
rewrite dprodE ?coprime_TIg -?cycleM ?consttC //.
by rewrite cent_cycle cycle_subG; apply/cent1P.
rewrite -(dprod_card (Ohm_dprod n def_x)) -(dprod_card (Mho_dprod n def_x)).
rewrite mulnCA -mulnA mulnCA mulnA.
rewrite !{}IHm ?(dprod_card def_x) ?(leq_trans _ lexm) {m lexm}//.
rewrite /order -(dprod_card def_x) -!orderE !order_constt ltn_Pmull //.
rewrite p_part -(expn0 p) ltn_exp2l 1?lognE ?prime_gt1 ?pdiv_prime //.
by rewrite order_gt0 pdiv_dvd.
rewrite proper_card // properEneq cycle_subG mem_cycle andbT.
by apply: contra (negbT p'x); move/eqP <-;
... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | mul_card_Ohm_Mho_abelian | |
grank_abelianG : abelian G -> 'm(G) = 'r(G).
Proof.
move=> cGG; apply/eqP; rewrite eqn_leq; apply/andP; split.
rewrite -size_abelian_type //; case/abelian_structure: cGG => b defG <-.
suffices <-: <<[set x in b]>> = G.
by rewrite (leq_trans (grank_min _)) // size_map cardsE card_size.
rewrite -{G defG}(bigdprodWY defG).
elim: b => [|x b IHb]; first by rewrite big_nil gen0.
by rewrite big_cons -joingE -joing_idr -IHb joing_idl joing_idr set_cons.
have [p p_pr ->] := rank_witness G; pose K := 'Mho^1(G).
have ->: 'r_p(G) = logn p #|G / K|.
rewrite p_rank_abelian // card_quotient /= ?gFnorm // -divgS ?Mho_sub //.
by rewrite -(mul_card_Ohm_Mho_abelian 1 cGG) mulnK ?cardG_gt0.
case: (grank_witness G) => B genB <-; rewrite -genB.
have{genB}: <<B>> \subset G by rewrite genB.
have [m] := ubnP #|B|; elim: m B => // m IHm B.
have [-> | [x Bx]] := set_0Vmem B; first by rewrite gen0 quotient1 cards1 logn1.
rewrite ltnS (cardsD1 x) Bx -[in <<B>>](setD1K Bx); set B' := B :\ x => ltB'm.
rewrite -joingE -joing_idl -joing_idr -/<[x]> join_subG => /andP[Gx sB'G].
rewrite cent_joinEl ?(sub_abelian_cent2 cGG) //.
have nKx: x \in 'N(K) by rewrite -cycle_subG (subset_trans Gx) ?gFnorm.
rewrite quotientMl ?cycle_subG // quotient_cycle //= -/K.
have le_Kxp_1: logn p #[coset K x] <= 1.
rewrite -(dvdn_Pexp2l _ _ (prime_gt1 p_pr)) -p_part -order_constt.
rewrite order_dvdn -morph_constt // -morphX ?groupX //= coset_id //.
by rewrite Mho_p_elt ?p_elt_constt ?groupX -?cycle_subG.
apply: leq_tran
... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | grank_abelian | |
rank_cycle(x : gT) : 'r(<[x]>) = (x != 1).
Proof.
have [->|ntx] := eqVneq x 1; first by rewrite cycle1 rank1.
apply/eqP; rewrite eqn_leq rank_gt0 cycle_eq1 ntx andbT.
by rewrite -grank_abelian ?cycle_abelian //= -(cards1 x) grank_min.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | rank_cycle | |
abelian_rank1_cyclicG : abelian G -> cyclic G = ('r(G) <= 1).
Proof.
move=> cGG; have [b defG atypG] := abelian_structure cGG.
apply/idP/idP; first by case/cyclicP=> x ->; rewrite rank_cycle leq_b1.
rewrite -size_abelian_type // -{}atypG -{}defG unlock.
by case: b => [|x []] //= _; rewrite ?cyclic1 // dprodg1 cycle_cyclic.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelian_rank1_cyclic | |
homocyclicA := abelian A && constant (abelian_type A). | Definition | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | homocyclic | |
homocyclic_Ohm_Mhon p G :
p.-group G -> homocyclic G -> 'Ohm_n(G) = 'Mho^(logn p (exponent G) - n)(G).
Proof.
move=> pG /andP[cGG homoG]; set e := exponent G.
have{pG} p_e: p.-nat e by apply: pnat_dvd pG; apply: exponent_dvdn.
have{homoG}: all (pred1 e) (abelian_type G).
move: homoG; rewrite /abelian_type -(prednK (cardG_gt0 G)) /=.
by case: (_ && _) (tag _); rewrite //= genGid eqxx.
have{cGG} [b defG <-] := abelian_structure cGG.
move: e => e in p_e *; elim: b => /= [|x b IHb] in G defG *.
by rewrite -defG big_nil (trivgP (Ohm_sub _ _)) (trivgP (Mho_sub _ _)).
case/andP=> /eqP ox e_b; rewrite big_cons in defG.
rewrite -(Ohm_dprod _ defG) -(Mho_dprod _ defG).
case/dprodP: defG => [[_ H _ defH] _ _ _]; rewrite defH (IHb H) //; congr (_ \x _).
by rewrite -ox in p_e *; rewrite (Ohm_p_cycle _ p_e) (Mho_p_cycle _ p_e).
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | homocyclic_Ohm_Mho | |
Ohm_Mho_homocyclic(n p : nat) G :
abelian G -> p.-group G -> 0 < n < logn p (exponent G) ->
'Ohm_n(G) = 'Mho^(logn p (exponent G) - n)(G) -> homocyclic G.
Proof.
set e := exponent G => cGG pG /andP[n_gt0 n_lte] eq_Ohm_Mho.
suffices: all (pred1 e) (abelian_type G).
by rewrite /homocyclic cGG; apply: all_pred1_constant.
case/abelian_structure: cGG (abelian_type_gt1 G) => b defG <-.
set H := G in defG eq_Ohm_Mho *; have sHG: H \subset G by [].
elim: b H defG sHG eq_Ohm_Mho => //= x b IHb H.
rewrite big_cons => defG; case/dprodP: defG (defG) => [[_ K _ defK]].
rewrite defK => defHm cxK; rewrite setIC => /trivgP-tiKx defHd.
rewrite -{}[in H \subset G]defHm mulG_subG cycle_subG ltnNge -trivg_card_le1.
case/andP=> Gx sKG; rewrite -(Mho_dprod _ defHd) => /esym defMho /andP[ntx ntb].
have{defHd} defOhm := Ohm_dprod n defHd.
apply/andP; split; last first.
apply: (IHb K) => //; have:= dprod_modr defMho (Mho_sub _ _).
rewrite -(dprod_modr defOhm (Ohm_sub _ _)).
rewrite !(trivgP (subset_trans (setIS _ _) tiKx)) ?Ohm_sub ?Mho_sub //.
by rewrite !dprod1g.
have:= dprod_modl defMho (Mho_sub _ _).
rewrite -(dprod_modl defOhm (Ohm_sub _ _)) .
rewrite !(trivgP (subset_trans (setSI _ _) tiKx)) ?Ohm_sub ?Mho_sub //.
move/eqP; rewrite eqEcard => /andP[_].
have p_x: p.-elt x := mem_p_elt pG Gx.
have [p_pr p_dv_x _] := pgroup_pdiv p_x ntx.
rewrite !dprodg1 (Ohm_p_cycle _ p_x) (Mho_p_cycle _ p_x) -!orderE.
rewrite orderXdiv ?leq_divLR ?pfactor_dvdn ?leq_subr //.
rewrite orderXgcd divn_mulAC ?dvdn_gcdl /
... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | Ohm_Mho_homocyclic | |
abelem_homocyclicp G : p.-abelem G -> homocyclic G.
Proof.
move=> abelG; have [_ cGG _] := and3P abelG.
rewrite /homocyclic cGG (@all_pred1_constant _ p) //.
case/abelian_structure: cGG (abelian_type_gt1 G) => b defG <- => b_gt1.
apply/allP=> _ /mapP[x b_x ->] /=; rewrite (abelem_order_p abelG) //.
rewrite -cycle_subG -(bigdprodWY defG) ?sub_gen //.
by rewrite bigcup_seq (bigcup_sup x).
by rewrite -order_gt1 [_ > 1](allP b_gt1) ?map_f.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelem_homocyclic | |
homocyclic1: homocyclic [1 gT].
Proof. exact: abelem_homocyclic (abelem1 _ 2). Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | homocyclic1 | |
Ohm1_homocyclicPp G : p.-group G -> abelian G ->
reflect ('Ohm_1(G) = 'Mho^(logn p (exponent G)).-1(G)) (homocyclic G).
Proof.
move=> pG cGG; set e := logn p (exponent G); rewrite -subn1.
apply: (iffP idP) => [homoG | ]; first exact: homocyclic_Ohm_Mho.
case: (ltnP 1 e) => [lt1e | ]; first exact: Ohm_Mho_homocyclic.
rewrite -subn_eq0 => /eqP->; rewrite Mho0 => <-.
exact: abelem_homocyclic (Ohm1_abelem pG cGG).
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | Ohm1_homocyclicP | |
abelian_type_homocyclicG :
homocyclic G -> abelian_type G = nseq 'r(G) (exponent G).
Proof.
case/andP=> cGG; rewrite -size_abelian_type // /abelian_type.
rewrite -(prednK (cardG_gt0 G)) /=; case: andP => //= _; move: (tag _) => H.
by move/all_pred1P->; rewrite genGid size_nseq.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelian_type_homocyclic | |
abelian_type_abelemp G : p.-abelem G -> abelian_type G = nseq 'r(G) p.
Proof.
move=> abelG; rewrite (abelian_type_homocyclic (abelem_homocyclic abelG)).
have [-> | ntG] := eqVneq G 1%G; first by rewrite rank1.
congr nseq; apply/eqP; rewrite eqn_dvd; have [pG _ ->] := and3P abelG.
have [p_pr] := pgroup_pdiv pG ntG; case/Cauchy=> // x Gx <- _.
exact: dvdn_exponent.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelian_type_abelem | |
max_card_abelianG :
abelian G -> #|G| <= exponent G ^ 'r(G) ?= iff homocyclic G.
Proof.
move=> cGG; have [b defG def_tG] := abelian_structure cGG.
have Gb: all [in G] b.
apply/allP=> x b_x; rewrite -(bigdprodWY defG); have [b1 b2] := splitPr b_x.
by rewrite big_cat big_cons /= mem_gen // setUCA inE cycle_id.
have ->: homocyclic G = all (pred1 (exponent G)) (abelian_type G).
rewrite /homocyclic cGG /abelian_type; case: #|G| => //= n.
by move: (_ (tag _)) => t; case: ifP => //= _; rewrite genGid eqxx.
rewrite -size_abelian_type // -{}def_tG -{defG}(bigdprod_card defG) size_map.
rewrite unlock; elim: b Gb => //= x b IHb; case/andP=> Gx Gb.
have eGgt0: exponent G > 0 := exponent_gt0 G.
have le_x_G: #[x] <= exponent G by rewrite dvdn_leq ?dvdn_exponent.
have:= leqif_mul (leqif_eq le_x_G) (IHb Gb).
by rewrite -expnS expn_eq0 eqn0Ngt eGgt0.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | max_card_abelian | |
card_homocyclicG : homocyclic G -> #|G| = (exponent G ^ 'r(G))%N.
Proof.
by move=> homG; have [cGG _] := andP homG; apply/eqP; rewrite max_card_abelian.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | card_homocyclic | |
abelian_type_dprod_homocyclicp K H G :
K \x H = G -> p.-group G -> homocyclic G ->
abelian_type K = nseq 'r(K) (exponent G)
/\ abelian_type H = nseq 'r(H) (exponent G).
Proof.
move=> defG pG homG; have [cGG _] := andP homG.
have /mulG_sub[sKG sHG]: K * H = G by case/dprodP: defG.
have [cKK cHH] := (abelianS sKG cGG, abelianS sHG cGG).
suffices: all (pred1 (exponent G)) (abelian_type K ++ abelian_type H).
rewrite all_cat => /andP[/all_pred1P-> /all_pred1P->].
by rewrite !size_abelian_type.
suffices def_atG: abelian_type K ++ abelian_type H =i abelian_type G.
rewrite (eq_all_r def_atG); apply/all_pred1P.
by rewrite size_abelian_type // -abelian_type_homocyclic.
have [bK defK atK] := abelian_structure cKK.
have [bH defH atH] := abelian_structure cHH.
apply/perm_mem; rewrite perm_sym -atK -atH -map_cat.
apply: (abelian_type_pgroup pG); first by rewrite big_cat defK defH.
have: all [pred m | m > 1] (map order (bK ++ bH)).
by rewrite map_cat all_cat atK atH !abelian_type_gt1.
by rewrite all_map (eq_all (@order_gt1 _)) all_predC has_pred1.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | abelian_type_dprod_homocyclic | |
dprod_homocyclicp K H G :
K \x H = G -> p.-group G -> homocyclic G -> homocyclic K /\ homocyclic H.
Proof.
move=> defG pG homG; have [cGG _] := andP homG.
have /mulG_sub[sKG sHG]: K * H = G by case/dprodP: defG.
have [abtK abtH] := abelian_type_dprod_homocyclic defG pG homG.
by rewrite /homocyclic !(abelianS _ cGG) // abtK abtH !constant_nseq.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | dprod_homocyclic | |
exponent_dprod_homocyclicp K H G :
K \x H = G -> p.-group G -> homocyclic G -> K :!=: 1 ->
exponent K = exponent G.
Proof.
move=> defG pG homG ntK; have [homK _] := dprod_homocyclic defG pG homG.
have [] := abelian_type_dprod_homocyclic defG pG homG.
by rewrite abelian_type_homocyclic // -['r(K)]prednK ?rank_gt0 => [[]|].
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | exponent_dprod_homocyclic | |
isog_abelian_typeG H : isog G H -> abelian_type G = abelian_type H.
Proof.
pose lnO p n gT (A : {set gT}) := logn p #|'Ohm_n.+1(A) : 'Ohm_n(A)|.
pose lni i p gT (A : {set gT}) := \max_(e < logn p #|A| | i < lnO p e _ A) e.+1.
suffices{G} nth_abty gT (G : {group gT}) i:
abelian G -> i < size (abelian_type G) ->
nth 1%N (abelian_type G) i = (\prod_(p < #|G|.+1) p ^ lni i p _ G)%N.
- move=> isoGH; case cGG: (abelian G); last first.
rewrite /abelian_type -(prednK (cardG_gt0 G)) -(prednK (cardG_gt0 H)) /=.
by rewrite {1}(genGid G) {1}(genGid H) -(isog_abelian isoGH) cGG.
have cHH: abelian H by rewrite -(isog_abelian isoGH).
have eq_sz: size (abelian_type G) = size (abelian_type H).
by rewrite !size_abelian_type ?(isog_rank isoGH).
apply: (@eq_from_nth _ 1%N) => // i lt_i_G; rewrite !nth_abty // -?eq_sz //.
rewrite /lni (card_isog isoGH); apply: eq_bigr => p _; congr (p ^ _)%N.
apply: eq_bigl => e; rewrite /lnO -!divgS ?(Ohm_leq _ (leqnSn _)) //=.
by have:= card_isog (gFisog _ isoGH) => /= eqF; rewrite !eqF.
move=> cGG.
have (p): path leq 0 (map (logn p) (rev (abelian_type G))).
move: (abelian_type_gt1 G) (abelian_type_dvdn_sorted G).
case: abelian_type => //= m t; rewrite rev_cons map_rcons.
elim: t m => //= n t IHt m /andP[/ltnW m_gt0 nt_gt1].
rewrite -cats1 cat_path rev_cons map_rcons last_rcons /=.
by case/andP=> /dvdn_leq_log-> // /IHt->.
have{cGG} [b defG <- b_sorted] := abelian_structure cGG.
rewrite size_map => ltib; rewrite (nth_map 1 _ _ ltib); set
... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | isog_abelian_type | |
eq_abelian_type_isogG H :
abelian G -> abelian H -> isog G H = (abelian_type G == abelian_type H).
Proof.
move=> cGG cHH; apply/idP/eqP; first exact: isog_abelian_type.
have{cGG} [bG defG <-] := abelian_structure cGG.
have{cHH} [bH defH <-] := abelian_structure cHH.
elim: bG bH G H defG defH => [|x bG IHb] [|y bH] // G H.
rewrite !big_nil => <- <- _.
by rewrite isog_cyclic_card ?cyclic1 ?cards1.
rewrite !big_cons => defG defH /= [eqxy eqb].
apply: (isog_dprod defG defH).
by rewrite isog_cyclic_card ?cycle_cyclic -?orderE ?eqxy /=.
case/dprodP: defG => [[_ G' _ defG]] _ _ _; rewrite defG.
case/dprodP: defH => [[_ H' _ defH]] _ _ _; rewrite defH.
exact: IHb eqb.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | eq_abelian_type_isog | |
isog_abelem_cardp G H :
p.-abelem G -> isog G H = p.-abelem H && (#|H| == #|G|).
Proof.
move=> abelG; apply/idP/andP=> [isoGH | [abelH eqGH]].
by rewrite -(isog_abelem isoGH) (card_isog isoGH).
rewrite eq_abelian_type_isog ?(@abelem_abelian _ p) //.
by rewrite !(@abelian_type_abelem _ p) ?(@rank_abelem _ p) // (eqP eqGH).
Qed.
Variables (D : {group aT}) (f : {morphism D >-> rT}). | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | isog_abelem_card | |
morphim_rank_abelianG : abelian G -> 'r(f @* G) <= 'r(G).
Proof.
move=> cGG; have sHG := subsetIr D G; apply: leq_trans (rankS sHG).
rewrite -!grank_abelian ?morphim_abelian ?(abelianS sHG) //=.
by rewrite -morphimIdom morphim_grank ?subsetIl.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | morphim_rank_abelian | |
morphim_p_rank_abelianp G : abelian G -> 'r_p(f @* G) <= 'r_p(G).
Proof.
move=> cGG; have sHG := subsetIr D G; apply: leq_trans (p_rankS p sHG).
have cHH := abelianS sHG cGG; rewrite -morphimIdom /=; set H := D :&: G.
have sylP := nilpotent_pcore_Hall p (abelian_nil cHH).
have sPH := pHall_sub sylP.
have sPD: 'O_p(H) \subset D by rewrite (subset_trans sPH) ?subsetIl.
rewrite -(p_rank_Sylow (morphim_pHall f sPD sylP)) -(p_rank_Sylow sylP) //.
rewrite -!rank_pgroup ?morphim_pgroup ?pcore_pgroup //.
by rewrite morphim_rank_abelian ?(abelianS sPH).
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | morphim_p_rank_abelian | |
isog_homocyclicG H : G \isog H -> homocyclic G = homocyclic H.
Proof.
move=> isoGH.
by rewrite /homocyclic (isog_abelian isoGH) (isog_abelian_type isoGH).
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | isog_homocyclic | |
quotient_rank_abelian: 'r(G / H) <= 'r(G).
Proof. exact: morphim_rank_abelian. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | quotient_rank_abelian | |
quotient_p_rank_abelian: 'r_p(G / H) <= 'r_p(G).
Proof. exact: morphim_p_rank_abelian. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | quotient_p_rank_abelian | |
fin_lmod_pchar_abelemp (R : nzRingType) (V : finLmodType R):
p \in [pchar R]%R -> p.-abelem [set: V].
Proof.
case/andP=> p_pr /eqP-pR0; apply/abelemP=> //.
by split=> [|v _]; rewrite ?zmod_abelian // zmodXgE -scaler_nat pR0 scale0r.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | fin_lmod_pchar_abelem | |
fin_Fp_lmod_abelemp (V : finLmodType 'F_p) :
prime p -> p.-abelem [set: V].
Proof. by move/pchar_Fp/fin_lmod_pchar_abelem->. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | fin_Fp_lmod_abelem | |
fin_ring_pchar_abelemp (R : finNzRingType) :
p \in [pchar R]%R -> p.-abelem [set: R].
Proof. exact: fin_lmod_pchar_abelem R^o. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | fin_ring_pchar_abelem | |
fin_lmod_char_abelem:= (fin_lmod_pchar_abelem) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use fin_ring_pchar_abelem instead.")] | Notation | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | fin_lmod_char_abelem | |
fin_ring_char_abelem:= (fin_ring_pchar_abelem) (only parsing). | Notation | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg count... | solvable/abelian.v | fin_ring_char_abelem | |
Definition_ := Finite_isGroup.Build bool addbA addFb addbb. | HB.instance | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | Definition | |
Sym: {set {perm T}} := setT. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | Sym | |
Sym_group:= Eval hnf in [group of Sym].
Local Notation "'Sym_T" := Sym. | Canonical | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | Sym_group | |
sign_morph:= @Morphism _ _ 'Sym_T _ (in2W (@odd_permM _)). | Canonical | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | sign_morph | |
Alt:= 'ker (@odd_perm T). | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | Alt | |
Alt_group:= Eval hnf in [group of Alt].
Local Notation "'Alt_T" := Alt. | Canonical | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | Alt_group | |
Alt_evenp : (p \in 'Alt_T) = ~~ p.
Proof. by rewrite !inE /=; case: odd_perm. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | Alt_even | |
Alt_subset: 'Alt_T \subset 'Sym_T.
Proof. exact: subsetT. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | Alt_subset | |
Alt_normal: 'Alt_T <| 'Sym_T.
Proof. exact: ker_normal. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | Alt_normal | |
Alt_norm: 'Sym_T \subset 'N('Alt_T).
Proof. by case/andP: Alt_normal. Qed.
Let n := #|T|. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | Alt_norm | |
Alt_index: 1 < n -> #|'Sym_T : 'Alt_T| = 2.
Proof.
move=> lt1n; rewrite -card_quotient ?Alt_norm //=.
have : ('Sym_T / 'Alt_T) \isog (@odd_perm T @* 'Sym_T) by apply: first_isog.
case/isogP=> g /injmP/card_in_imset <-.
rewrite /morphim setIid=> ->; rewrite -card_bool; apply: eq_card => b.
apply/imsetP; case: b => /=; last first.
by exists (1 : {perm T}); [rewrite setIid inE | rewrite odd_perm1].
case: (pickP T) lt1n => [x1 _ | d0]; last by rewrite /n eq_card0.
rewrite /n (cardD1 x1) ltnS lt0n => /existsP[x2 /=].
by rewrite eq_sym andbT -odd_tperm; exists (tperm x1 x2); rewrite ?inE.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | Alt_index | |
card_Sym: #|'Sym_T| = n`!.
Proof.
rewrite -[n]cardsE -card_perm; apply: eq_card => p.
by apply/idP/subsetP=> [? ?|]; rewrite !inE.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | card_Sym | |
card_Alt: 1 < n -> (2 * #|'Alt_T|)%N = n`!.
Proof.
by move/Alt_index <-; rewrite mulnC (Lagrange Alt_subset) card_Sym.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | card_Alt | |
Sym_trans: [transitive^n 'Sym_T, on setT | 'P].
Proof.
apply/imsetP; pose t1 := [tuple of enum T].
have dt1: t1 \in n.-dtuple(setT) by rewrite inE enum_uniq; apply/subsetP.
exists t1 => //; apply/setP=> t; apply/idP/imsetP=> [|[a _ ->{t}]]; last first.
by apply: n_act_dtuple => //; apply/astabsP=> x; rewrite !inE.
case/dtuple_onP=> injt _; have injf := inj_comp injt enum_rank_inj.
exists (perm injf); first by rewrite inE.
apply: eq_from_tnth => i; rewrite tnth_map /= [aperm _ _]permE; congr tnth.
by rewrite (tnth_nth (enum_default i)) enum_valK.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | Sym_trans | |
Alt_trans: [transitive^n.-2 'Alt_T, on setT | 'P].
Proof.
case n_m2: n Sym_trans => [|[|m]] /= tr_m2; try exact: ntransitive0.
have tr_m := ntransitive_weak (leqW (leqnSn m)) tr_m2.
case/imsetP: tr_m2; case/tupleP=> x; case/tupleP=> y t.
rewrite !dtuple_on_add 2![x \in _]inE inE negb_or /= -!andbA.
case/and4P=> nxy ntx nty dt _; apply/imsetP; exists t => //; apply/setP=> u.
apply/idP/imsetP=> [|[a _ ->{u}]]; last first.
by apply: n_act_dtuple => //; apply/astabsP=> z; rewrite !inE.
case/(atransP2 tr_m dt)=> /= a _ ->{u}.
case odd_a: (odd_perm a); last by exists a => //; rewrite !inE /= odd_a.
exists (tperm x y * a); first by rewrite !inE /= odd_permM odd_tperm nxy odd_a.
apply/val_inj/eq_in_map => z tz; rewrite actM /= /aperm; congr (a _).
by case: tpermP ntx nty => // <-; rewrite tz.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | Alt_trans | |
aperm_faithful(A : {group {perm T}}) : [faithful A, on setT | 'P].
Proof.
by apply/faithfulP=> /= p _ np1; apply/eqP/perm_act1P=> y; rewrite np1 ?inE.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | aperm_faithful | |
trivial_Alt_2(T : finType) : #|T| <= 2 -> 'Alt_T = 1.
Proof.
rewrite leq_eqVlt => /predU1P[] oT.
by apply: card_le1_trivg; rewrite -leq_double -mul2n card_Alt oT.
suffices Sym1: 'Sym_T = 1 by apply/trivgP; rewrite -Sym1 subsetT.
by apply: card1_trivg; rewrite card_Sym; case: #|T| oT; do 2?case.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | trivial_Alt_2 | |
simple_Alt_3(T : finType) : #|T| = 3 -> simple 'Alt_T.
Proof.
move=> T3; have{T3} oA: #|'Alt_T| = 3.
by apply: double_inj; rewrite -mul2n card_Alt T3.
apply/simpleP; split=> [|K]; [by rewrite trivg_card1 oA | case/andP=> sKH _].
have:= cardSg sKH; rewrite oA dvdn_divisors // !inE orbC /= -oA.
case/pred2P=> eqK; [right | left]; apply/eqP.
by rewrite eqEcard sKH eqK leqnn.
by rewrite eq_sym eqEcard sub1G eqK cards1.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | simple_Alt_3 | |
not_simple_Alt_4(T : finType) : #|T| = 4 -> ~~ simple 'Alt_T.
Proof.
move=> oT; set A := 'Alt_T.
have oA: #|A| = 12 by apply: double_inj; rewrite -mul2n card_Alt oT.
suffices [p]: exists p, [/\ prime p, 1 < #|A|`_p < #|A| & #|'Syl_p(A)| == 1%N].
case=> p_pr pA_int; rewrite /A; case/normal_sylowP=> P; case/pHallP.
rewrite /= -/A => sPA pP nPA; apply/simpleP=> [] [_]; rewrite -pP in pA_int.
by case/(_ P)=> // defP; rewrite defP oA ?cards1 in pA_int.
have: #|'Syl_3(A)| \in filter [pred d | d %% 3 == 1%N] (divisors 12).
by rewrite mem_filter -dvdn_divisors //= -oA card_Syl_dvd ?card_Syl_mod.
rewrite /= oA mem_seq2 orbC.
case/predU1P=> [oQ3|]; [exists 2 | exists 3]; split; rewrite ?p_part //.
pose A3 := [set x : {perm T} | #[x] == 3]; suffices oA3: #|A :&: A3| = 8.
have sQ2 P: P \in 'Syl_2(A) -> P :=: A :\: A3.
rewrite inE pHallE oA p_part -natTrecE /= => /andP[sPA /eqP oP].
apply/eqP; rewrite eqEcard -(leq_add2l 8) -{1}oA3 cardsID oA oP.
rewrite andbT subsetD sPA; apply/exists_inP=> -[x] /= Px.
by rewrite inE => /eqP ox; have:= order_dvdG Px; rewrite oP ox.
have [/= P sylP] := Sylow_exists 2 [group of A].
rewrite -(([set P] =P 'Syl_2(A)) _) ?cards1 // eqEsubset sub1set inE sylP.
by apply/subsetP=> Q sylQ; rewrite inE -val_eqE /= !sQ2 // inE.
rewrite -[8]/(4 * 2)%N -{}oQ3 -sum1_card -sum_nat_const.
rewrite (partition_big (fun x => <[x]>%G) [in 'Syl_3(A)]) => [|x]; last first.
by case/setIP=> Ax; rewrite /= !inE pHallE p_part cycle_subG Ax oA.
apply: eq_bigr => Q;
... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | not_simple_Alt_4 | |
simple_Alt5_base(T : finType) : #|T| = 5 -> simple 'Alt_T.
Proof.
move=> oT.
have F1: #|'Alt_T| = 60 by apply: double_inj; rewrite -mul2n card_Alt oT.
have FF (H : {group {perm T}}): H <| 'Alt_T -> H :<>: 1 -> 20 %| #|H|.
- move=> Hh1 Hh3.
have [x _]: exists x, x \in T by apply/existsP/eqP; rewrite oT.
have F2 := Alt_trans T; rewrite oT /= in F2.
have F3: [transitive 'Alt_T, on setT | 'P] by apply: ntransitive1 F2.
have F4: [primitive 'Alt_T, on setT | 'P] by apply: ntransitive_primitive F2.
case: (prim_trans_norm F4 Hh1) => F5.
by case: Hh3; apply/trivgP; apply: subset_trans F5 (aperm_faithful _).
have F6: 5 %| #|H| by rewrite -oT -cardsT (atrans_dvd F5).
have F7: 4 %| #|H|.
have F7: #|[set~ x]| = 4 by rewrite cardsC1 oT.
case: (pickP [in [set~ x]]) => [y Hy | ?]; last by rewrite eq_card0 in F7.
pose K := 'C_H[x | 'P]%G.
have F8 : K \subset H by apply: subsetIl.
pose Gx := 'C_('Alt_T)[x | 'P]%G.
have F9: [transitive^2 Gx, on [set~ x] | 'P].
by rewrite -[[set~ x]]setTI -setDE stab_ntransitive ?inE.
have F10: [transitive Gx, on [set~ x] | 'P].
exact: ntransitive1 F9.
have F11: [primitive Gx, on [set~ x] | 'P].
exact: ntransitive_primitive F9.
have F12: K \subset Gx by apply: setSI; apply: normal_sub.
have F13: K <| Gx by rewrite /(K <| _) F12 normsIG // normal_norm.
case: (prim_trans_norm F11 F13) => Ksub; last first.
by apply: dvdn_trans (cardSg F8); rewrite -F7; apply: atrans_dvd Ksub.
have F14: [faith
... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | simple_Alt5_base | |
T':= {y | y != x}. | Notation | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | T' | |
rfd_funP(p : {perm T}) (u : T') :
let p1 := if p x == x then p else 1 in p1 (val u) != x.
Proof.
case: (p x =P x) => /= [pxx | _]; last by rewrite perm1 (valP u).
by rewrite -[x in _ != x]pxx (inj_eq perm_inj); apply: (valP u).
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | rfd_funP | |
rfd_funp := [fun u => Sub ((_ : {perm T}) _) (rfd_funP p u) : T']. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | rfd_fun | |
rfdPp : injective (rfd_fun p).
Proof.
apply: can_inj (rfd_fun p^-1) _ => u; apply: val_inj => /=.
rewrite -(can_eq (permK p)) permKV eq_sym.
by case: eqP => _; rewrite !(perm1, permK).
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | rfdP | |
rfdp := perm (@rfdP p).
Hypothesis card_T : 2 < #|T|. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | rfd | |
rfd_morph: {in 'C_('Sym_T)[x | 'P] &, {morph rfd : y z / y * z}}.
Proof.
move=> p q; rewrite !setIA !setIid; move/astab1P=> p_x; move/astab1P=> q_x.
apply/permP=> u; apply: val_inj.
by rewrite permE /= !permM !permE /= [p x]p_x [q x]q_x eqxx permM /=.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | rfd_morph | |
rfd_morphism:= Morphism rfd_morph. | Canonical | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | rfd_morphism | |
rgd_fun(p : {perm T'}) :=
[fun x1 => if insub x1 is Some u then sval (p u) else x]. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | rgd_fun | |
rgdPp : injective (rgd_fun p).
Proof.
apply: can_inj (rgd_fun p^-1) _ => y /=.
case: (insubP _ y) => [u _ val_u|]; first by rewrite valK permK.
by rewrite negbK; move/eqP->; rewrite insubF //= eqxx.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | rgdP | |
rgdp := perm (@rgdP p). | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | rgd | |
rfd_odd(p : {perm T}) : p x = x -> rfd p = p :> bool.
Proof.
have rfd1: rfd 1 = 1.
by apply/permP => u; apply: val_inj; rewrite permE /= if_same !perm1.
have [n] := ubnP #|[set x | p x != x]|; elim: n p => // n IHn p le_p_n px_x.
have [p_id | [x1 Hx1]] := set_0Vmem [set x | p x != x].
suffices ->: p = 1 by rewrite rfd1 !odd_perm1.
by apply/permP => z; apply: contraFeq (in_set0 z); rewrite perm1 -p_id inE.
have nx1x: x1 != x by apply: contraTneq Hx1 => ->; rewrite inE px_x eqxx.
have npxx1: p x != x1 by apply: contraNneq nx1x => <-; rewrite px_x.
have npx1x: p x1 != x by rewrite -px_x (inj_eq perm_inj).
pose p1 := p * tperm x1 (p x1).
have fix_p1 y: p y = y -> p1 y = y.
by move=> pyy; rewrite permM; have [<-|/perm_inj<-|] := tpermP; rewrite ?pyy.
have p1x_x: p1 x = x by apply: fix_p1.
have{le_p_n} lt_p1_n: #|[set x | p1 x != x]| < n.
move: le_p_n; rewrite ltnS (cardsD1 x1) Hx1; apply/leq_trans/subset_leq_card.
rewrite subsetD1 inE permM tpermR eqxx andbT.
by apply/subsetP=> y /[!inE]; apply: contraNneq=> /fix_p1->.
transitivity (p1 (+) true); last first.
by rewrite odd_permM odd_tperm -Hx1 inE eq_sym addbK.
have ->: p = p1 * tperm x1 (p x1) by rewrite -tpermV mulgK.
rewrite morphM; last 2 first; first by rewrite 2!inE; apply/astab1P.
by rewrite 2!inE; apply/astab1P; rewrite -[RHS]p1x_x permM px_x.
rewrite odd_permM IHn //=; congr (_ (+) _).
pose x2 : T' := Sub x1 nx1x; pose px2 : T' := Sub (p x1) npx1x.
suffices ->: rfd (tperm x1 (p x1)) = tperm x2 px2.
by rewrite
... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | rfd_odd | |
rfd_iso: 'C_('Alt_T)[x | 'P] \isog 'Alt_T'.
Proof.
have rgd_x p: rgd p x = x by rewrite permE /= insubF //= eqxx.
have rfd_rgd p: rfd (rgd p) = p.
apply/permP => [[z Hz]]; apply/val_eqP; rewrite !permE.
by rewrite /= [rgd _ _]permE /= insubF eqxx // permE /= insubT.
have sSd: 'C_('Alt_T)[x | 'P] \subset 'dom rfd.
by apply/subsetP=> p /[!inE]/= /andP[].
apply/isogP; exists [morphism of restrm sSd rfd] => /=; last first.
rewrite morphim_restrm setIid; apply/setP=> z; apply/morphimP/idP=> [[p _]|].
case/setIP; rewrite Alt_even => Hp; move/astab1P=> Hp1 ->.
by rewrite Alt_even rfd_odd.
have dz': rgd z x == x by rewrite rgd_x.
move=> kz; exists (rgd z); last by rewrite /= rfd_rgd.
by rewrite 2!inE (sameP astab1P eqP).
rewrite 4!inE /= (sameP astab1P eqP) dz' -rfd_odd; last exact/eqP.
by rewrite rfd_rgd mker // ?set11.
apply/injmP=> x1 y1 /=.
case/setIP=> Hax1; move/astab1P; rewrite /= /aperm => Hx1.
case/setIP=> Hay1; move/astab1P; rewrite /= /aperm => Hy1 Hr.
apply/permP => z.
case (z =P x) => [->|]; [by rewrite Hx1 | move/eqP => nzx].
move: (congr1 (fun q : {perm T'} => q (Sub z nzx)) Hr).
by rewrite !permE => [[]]; rewrite Hx1 Hy1 !eqxx.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | rfd_iso | |
simple_Alt5(T : finType) : #|T| >= 5 -> simple 'Alt_T.
Proof.
suff F1 n: #|T| = n + 5 -> simple 'Alt_T by move/subnK/esym/F1.
elim: n T => [| n Hrec T Hde]; first exact: simple_Alt5_base.
have oT: 5 < #|T| by rewrite Hde addnC.
apply/simpleP; split=> [|H Hnorm]; last have [Hh1 nH] := andP Hnorm.
rewrite trivg_card1 -[#|_|]half_double -mul2n card_Alt Hde addnC //.
by rewrite addSn factS mulnC -(prednK (fact_gt0 _)).
case E1: (pred0b T); first by rewrite /pred0b in E1; rewrite (eqP E1) in oT.
case/pred0Pn: E1 => x _; have Hx := in_setT x.
have F2: [transitive^4 'Alt_T, on setT | 'P].
by apply: ntransitive_weak (Alt_trans T); rewrite -(subnKC oT).
have F3 := ntransitive1 (isT: 0 < 4) F2.
have F4 := ntransitive_primitive (isT: 1 < 4) F2.
case Hcard1: (#|H| == 1%N); move/eqP: Hcard1 => Hcard1.
by left; apply: card1_trivg; rewrite Hcard1.
right; case: (prim_trans_norm F4 Hnorm) => F5.
by rewrite (trivGP (subset_trans F5 (aperm_faithful _))) cards1 in Hcard1.
case E1: (pred0b (predD1 T x)).
rewrite /pred0b in E1; move: oT.
by rewrite (cardD1 x) (eqP E1); case: (T x).
case/pred0Pn: E1 => y Hdy; case/andP: (Hdy) => diff_x_y Hy.
pose K := 'C_H[x | 'P]%G.
have F8: K \subset H by apply: subsetIl.
pose Gx := 'C_('Alt_T)[x | 'P].
have F9: [transitive^3 Gx, on [set~ x] | 'P].
by rewrite -[[set~ x]]setTI -setDE stab_ntransitive ?inE.
have F10: [transitive Gx, on [set~ x] | 'P].
by apply: ntransitive1 F9.
have F11: [primitive Gx, on [set~ x] | 'P].
by apply: ntransitive_primitive F9.
... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | simple_Alt5 | |
gen_tperm_circular_shift(X : finType) x y c : prime #|X| ->
x != y -> #[c]%g = #|X| ->
<<[set tperm x y; c]>>%g = ('Sym_X)%g.
Proof.
move=> Xprime neq_xy ord_c; apply/eqP; rewrite eqEsubset subsetT/=.
have c_gt1 : (1 < #[c]%g)%N by rewrite ord_c prime_gt1.
have cppSS : #[c]%g.-2.+2 = #|X| by rewrite ?prednK ?ltn_predRL.
pose f (i : 'Z_#[c]%g) : X := Zpm i x.
have [g fK gK] : bijective f.
apply: inj_card_bij; rewrite ?cppSS ?card_ord// /f /Zpm => i j cijx.
pose stabx := ('C_<[c]>[x | 'P])%g.
have cjix : (c ^+ (j - i)%R)%g x = x.
by apply: (@perm_inj _ (c ^+ i)%g); rewrite -permM -expgD_Zp// addrNK.
have : (c ^+ (j - i)%R)%g \in stabx.
by rewrite !inE ?groupX ?mem_gen ?sub1set ?inE// ['P%act _ _]cjix eqxx.
rewrite [stabx]perm_prime_astab// => /set1gP.
move=> /(congr1 ( *%g (c ^+ i))); rewrite -expgD_Zp// addrC addrNK mulg1.
by move=> /eqP; rewrite eq_expg_ord// ?cppSS ?ord_c// => /eqP->.
pose gsf s := g \o s \o f.
have gsf_inj (s : {perm X}) : injective (gsf s).
apply: inj_comp; last exact: can_inj fK.
by apply: inj_comp; [exact: can_inj gK|exact: perm_inj].
pose fsg s := f \o s \o g.
have fsg_inj (s : {perm _}) : injective (fsg s).
apply: inj_comp; last exact: can_inj gK.
by apply: inj_comp; [exact: can_inj fK|exact: perm_inj].
have gsf_morphic : morphic 'Sym_X (fun s => perm (gsf_inj s)).
apply/morphicP => u v _ _; apply/permP => /= i.
by rewrite !permE/= !permE /gsf /= gK permM.
pose phi := morphm gsf_morphic; rewrite /= in phi.
have phi_inj : ('injm phi)%g.
... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require I... | solvable/alt.v | gen_tperm_circular_shift |
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