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 |
|---|---|---|---|---|---|---|
p'groupEpip G : p^'.-group G = (p \notin \pi(G)).
Proof. exact: p'natEpi (cardG_gt0 G). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
p'groupEpi
| |
pgroup_piG : \pi(G).-group G.
Proof. by rewrite /=; apply: pnat_pi. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pgroup_pi
| |
partG_eq1pi G : (#|G|`_pi == 1)%N = pi^'.-group G.
Proof. exact: partn_eq1 (cardG_gt0 G). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
partG_eq1
| |
pgroupPpi G :
reflect (forall p, prime p -> p %| #|G| -> p \in pi) (pi.-group G).
Proof. exact: pnatP. Qed.
Arguments pgroupP {pi G}.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pgroupP
| |
pgroup1pi : pi.-group [1 gT].
Proof. by rewrite /pgroup cards1. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pgroup1
| |
pgroupSpi G H : H \subset G -> pi.-group G -> pi.-group H.
Proof. by move=> sHG; apply: pnat_dvd (cardSg sHG). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pgroupS
| |
oddSgG H : H \subset G -> odd #|G| -> odd #|H|.
Proof. by rewrite !odd_2'nat; apply: pgroupS. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
oddSg
| |
odd_pgroup_oddp G : odd p -> p.-group G -> odd #|G|.
Proof.
move=> p_odd pG; rewrite odd_2'nat (pi_pnat pG) // !inE.
by case: eqP p_odd => // ->.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
odd_pgroup_odd
| |
card_pgroupp G : p.-group G -> #|G| = (p ^ logn p #|G|)%N.
Proof. by move=> pG; rewrite -p_part part_pnat_id. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
card_pgroup
| |
properG_ltn_logp G H :
p.-group G -> H \proper G -> logn p #|H| < logn p #|G|.
Proof.
move=> pG; rewrite properEneq eqEcard andbC ltnNge => /andP[sHG].
rewrite sHG /= {1}(card_pgroup pG) {1}(card_pgroup (pgroupS sHG pG)).
by apply: contra; case: p {pG} => [|p] leHG; rewrite ?logn0 // leq_pexp2l.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
properG_ltn_log
| |
pgroupMpi G H : pi.-group (G * H) = pi.-group G && pi.-group H.
Proof.
have GH_gt0: 0 < #|G :&: H| := cardG_gt0 _.
rewrite /pgroup -(mulnK #|_| GH_gt0) -mul_cardG -(LagrangeI G H) -mulnA.
by rewrite mulKn // -(LagrangeI H G) setIC !pnatM andbCA; case: (pnat _).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pgroupM
| |
pgroupJpi G x : pi.-group (G :^ x) = pi.-group G.
Proof. by rewrite /pgroup cardJg. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pgroupJ
| |
pgroup_pp P : p.-group P -> p_group P.
Proof.
case: (leqP #|P| 1); first by move=> /card_le1_trivg-> _; apply: pgroup1.
move/pdiv_prime=> pr_q pgP; have:= pgroupP pgP _ pr_q (pdiv_dvd _).
by rewrite /p_group => /eqnP->.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pgroup_p
| |
p_groupPP : p_group P -> exists2 p, prime p & p.-group P.
Proof.
case: (ltnP 1 #|P|); first by move/pdiv_prime; exists (pdiv #|P|).
by move/card_le1_trivg=> -> _; exists 2 => //; apply: pgroup1.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
p_groupP
| |
pgroup_pdivp G :
p.-group G -> G :!=: 1 ->
[/\ prime p, p %| #|G| & exists m, #|G| = p ^ m.+1]%N.
Proof.
move=> pG; rewrite trivg_card1; case/p_groupP: (pgroup_p pG) => q q_pr qG.
move/implyP: (pgroupP pG q q_pr); case/p_natP: qG => // [[|m] ->] //.
by rewrite dvdn_exp // => /eqnP <- _; split; rewrite ?dvdn_exp //; exists m.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pgroup_pdiv
| |
coprime_p'groupp K R :
coprime #|K| #|R| -> p.-group R -> R :!=: 1 -> p^'.-group K.
Proof.
move=> coKR pR ntR; have [p_pr _ [e oK]] := pgroup_pdiv pR ntR.
by rewrite oK coprime_sym coprime_pexpl // prime_coprime // -p'natE in coKR.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
coprime_p'group
| |
card_Hallpi G H : pi.-Hall(G) H -> #|H| = (#|G|`_pi)%N.
Proof.
case/and3P=> sHG piH pi'H; rewrite -(Lagrange sHG).
by rewrite partnM ?Lagrange // part_pnat_id ?part_p'nat ?muln1.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
card_Hall
| |
pHall_subpi A B : pi.-Hall(A) B -> B \subset A.
Proof. by case/andP. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pHall_sub
| |
pHall_pgrouppi A B : pi.-Hall(A) B -> pi.-group B.
Proof. by case/and3P. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pHall_pgroup
| |
pHallPpi G H : reflect (H \subset G /\ #|H| = #|G|`_pi)%N (pi.-Hall(G) H).
Proof.
apply: (iffP idP) => [piH | [sHG oH]].
by split; [apply: pHall_sub piH | apply: card_Hall].
rewrite /pHall sHG -divgS // /pgroup oH.
by rewrite -{2}(@partnC pi #|G|) ?mulKn ?part_pnat.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pHallP
| |
pHallEpi G H : pi.-Hall(G) H = (H \subset G) && (#|H| == #|G|`_pi)%N.
Proof. by apply/pHallP/andP=> [] [->] /eqP. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pHallE
| |
coprime_mulpG_Hallpi G K R :
K * R = G -> pi.-group K -> pi^'.-group R ->
pi.-Hall(G) K /\ pi^'.-Hall(G) R.
Proof.
move=> defG piK pi'R; apply/andP.
rewrite /pHall piK -!divgS /= -defG ?mulG_subl ?mulg_subr //= pnatNK.
by rewrite coprime_cardMg ?(pnat_coprime piK) // mulKn ?mulnK //; apply/and3P.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
coprime_mulpG_Hall
| |
coprime_mulGp_Hallpi G K R :
K * R = G -> pi^'.-group K -> pi.-group R ->
pi^'.-Hall(G) K /\ pi.-Hall(G) R.
Proof.
move=> defG pi'K piR; apply/andP; rewrite andbC; apply/andP.
by apply: coprime_mulpG_Hall => //; rewrite -(comm_group_setP _) defG ?groupP.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
coprime_mulGp_Hall
| |
eq_in_pHallpi rho G H :
{in \pi(G), pi =i rho} -> pi.-Hall(G) H = rho.-Hall(G) H.
Proof.
move=> eq_pi_rho; apply: andb_id2l => sHG.
congr (_ && _); apply: eq_in_pnat => p piHp.
by apply: eq_pi_rho; apply: (piSg sHG).
by congr (~~ _); apply: eq_pi_rho; apply: (pi_of_dvd (dvdn_indexg G H)).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
eq_in_pHall
| |
eq_pHallpi rho G H : pi =i rho -> pi.-Hall(G) H = rho.-Hall(G) H.
Proof. by move=> eq_pi_rho; apply: eq_in_pHall (in1W eq_pi_rho). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
eq_pHall
| |
eq_p'Hallpi rho G H : pi =i rho -> pi^'.-Hall(G) H = rho^'.-Hall(G) H.
Proof. by move=> eq_pi_rho; apply: eq_pHall (eq_negn _). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
eq_p'Hall
| |
pHallNKpi G H : pi^'^'.-Hall(G) H = pi.-Hall(G) H.
Proof. exact: eq_pHall (negnK _). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pHallNK
| |
subHall_Hallpi rho G H K :
rho.-Hall(G) H -> {subset pi <= rho} -> pi.-Hall(H) K -> pi.-Hall(G) K.
Proof.
move=> hallH pi_sub_rho hallK.
rewrite pHallE (subset_trans (pHall_sub hallK) (pHall_sub hallH)) /=.
by rewrite (card_Hall hallK) (card_Hall hallH) partn_part.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
subHall_Hall
| |
subHall_Sylowpi p G H P :
pi.-Hall(G) H -> p \in pi -> p.-Sylow(H) P -> p.-Sylow(G) P.
Proof.
move=> hallH pi_p sylP; have [sHG piH _] := and3P hallH.
rewrite pHallE (subset_trans (pHall_sub sylP) sHG) /=.
by rewrite (card_Hall sylP) (card_Hall hallH) partn_part // => q; move/eqnP->.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
subHall_Sylow
| |
pHall_Hallpi A B : pi.-Hall(A) B -> Hall A B.
Proof. by case/and3P=> sBA piB pi'B; rewrite /Hall sBA (pnat_coprime piB). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pHall_Hall
| |
Hall_piG H : Hall G H -> \pi(H).-Hall(G) H.
Proof.
by case/andP=> sHG coHG /=; rewrite /pHall sHG /pgroup pnat_pi -?coprime_pi'.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
Hall_pi
| |
HallPG H : Hall G H -> exists pi, pi.-Hall(G) H.
Proof. by exists \pi(H); apply: Hall_pi. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
HallP
| |
sdprod_HallG K H : K ><| H = G -> Hall G K = Hall G H.
Proof.
case/sdprod_context=> /andP[sKG _] sHG defG _ tiKH.
by rewrite /Hall sKG sHG -!divgS // -defG TI_cardMg // coprime_sym mulKn ?mulnK.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
sdprod_Hall
| |
coprime_sdprod_Hall_lG K H : K ><| H = G -> coprime #|K| #|H| = Hall G K.
Proof.
case/sdprod_context=> /andP[sKG _] _ defG _ tiKH.
by rewrite /Hall sKG -divgS // -defG TI_cardMg ?mulKn.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
coprime_sdprod_Hall_l
| |
coprime_sdprod_Hall_rG K H : K ><| H = G -> coprime #|K| #|H| = Hall G H.
Proof.
by move=> defG; rewrite (coprime_sdprod_Hall_l defG) (sdprod_Hall defG).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
coprime_sdprod_Hall_r
| |
compl_pHallpi K H G :
pi.-Hall(G) K -> (H \in [complements to K in G]) = pi^'.-Hall(G) H.
Proof.
move=> hallK; apply/complP/idP=> [[tiKH mulKH] | hallH].
have [_] := andP hallK; rewrite /pHall pnatNK -{3}(invGid G) -mulKH mulG_subr.
rewrite invMG !indexMg -indexgI andbC.
by rewrite -[#|K : H|]indexgI setIC tiKH !indexg1.
have [[sKG piK _] [sHG pi'H _]] := (and3P hallK, and3P hallH).
have tiKH: K :&: H = 1 := coprime_TIg (pnat_coprime piK pi'H).
split=> //; apply/eqP; rewrite eqEcard mul_subG //= TI_cardMg //.
by rewrite (card_Hall hallK) (card_Hall hallH) partnC.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
compl_pHall
| |
compl_p'Hallpi K H G :
pi^'.-Hall(G) K -> (H \in [complements to K in G]) = pi.-Hall(G) H.
Proof. by move/compl_pHall->; apply: eq_pHall (negnK pi). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
compl_p'Hall
| |
sdprod_normal_p'HallPpi K H G :
K <| G -> pi^'.-Hall(G) H -> reflect (K ><| H = G) (pi.-Hall(G) K).
Proof.
move=> nsKG hallH; rewrite -(compl_p'Hall K hallH).
exact: sdprod_normal_complP.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
sdprod_normal_p'HallP
| |
sdprod_normal_pHallPpi K H G :
K <| G -> pi.-Hall(G) H -> reflect (K ><| H = G) (pi^'.-Hall(G) K).
Proof.
by move=> nsKG hallH; apply: sdprod_normal_p'HallP; rewrite ?pHallNK.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
sdprod_normal_pHallP
| |
pHallJ2pi G H x : pi.-Hall(G :^ x) (H :^ x) = pi.-Hall(G) H.
Proof. by rewrite !pHallE conjSg !cardJg. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pHallJ2
| |
pHallJnormpi G H x : x \in 'N(G) -> pi.-Hall(G) (H :^ x) = pi.-Hall(G) H.
Proof. by move=> Nx; rewrite -{1}(normP Nx) pHallJ2. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pHallJnorm
| |
pHallJpi G H x : x \in G -> pi.-Hall(G) (H :^ x) = pi.-Hall(G) H.
Proof. by move=> Gx; rewrite -{1}(conjGid Gx) pHallJ2. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pHallJ
| |
HallJG H x : x \in G -> Hall G (H :^ x) = Hall G H.
Proof.
by move=> Gx; rewrite /Hall -!divgI -{1 3}(conjGid Gx) conjSg -conjIg !cardJg.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
HallJ
| |
psubgroupJpi G H x :
x \in G -> pi.-subgroup(G) (H :^ x) = pi.-subgroup(G) H.
Proof. by move=> Gx; rewrite /psubgroup pgroupJ -{1}(conjGid Gx) conjSg. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
psubgroupJ
| |
p_groupJP x : p_group (P :^ x) = p_group P.
Proof. by rewrite /p_group cardJg pgroupJ. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
p_groupJ
| |
SylowJG P x : x \in G -> Sylow G (P :^ x) = Sylow G P.
Proof. by move=> Gx; rewrite /Sylow p_groupJ HallJ. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
SylowJ
| |
p_Sylowp G P : p.-Sylow(G) P -> Sylow G P.
Proof.
by move=> pP; rewrite /Sylow (pgroup_p (pHall_pgroup pP)) (pHall_Hall pP).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
p_Sylow
| |
pHall_sublpi G K H :
H \subset K -> K \subset G -> pi.-Hall(G) H -> pi.-Hall(K) H.
Proof.
by move=> sHK sKG; rewrite /pHall sHK => /and3P[_ ->]; apply/pnat_dvd/indexSg.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pHall_subl
| |
Hall1G : Hall G 1.
Proof. by rewrite /Hall sub1G cards1 coprime1n. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
Hall1
| |
p_group1: @p_group gT 1.
Proof. by rewrite (@pgroup_p 2) ?pgroup1. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
p_group1
| |
Sylow1G : Sylow G 1.
Proof. by rewrite /Sylow p_group1 Hall1. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
Sylow1
| |
SylowPG P : reflect (exists2 p, prime p & p.-Sylow(G) P) (Sylow G P).
Proof.
apply: (iffP idP) => [| [p _]]; last exact: p_Sylow.
case/andP=> /p_groupP[p p_pr] /p_natP[[P1 _ | n oP /Hall_pi]]; last first.
by rewrite /= oP pi_of_exp // (eq_pHall _ _ (pi_of_prime _)) //; exists p.
have{p p_pr P1} ->: P :=: 1 by apply: card1_trivg; rewrite P1.
pose p := pdiv #|G|.+1; have p_pr: prime p by rewrite pdiv_prime ?ltnS.
exists p; rewrite // pHallE sub1G cards1 part_p'nat //.
apply/pgroupP=> q pr_q qG; apply/eqnP=> def_q.
have: p %| #|G| + 1 by rewrite addn1 pdiv_dvd.
by rewrite dvdn_addr -def_q // Euclid_dvd1.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
SylowP
| |
p_elt_exppi x m : pi.-elt (x ^+ m) = (#[x]`_pi^' %| m).
Proof.
apply/idP/idP=> [pi_xm | /dvdnP[q ->{m}]]; last first.
rewrite mulnC; apply: pnat_dvd (part_pnat pi #[x]).
by rewrite order_dvdn -expgM mulnC mulnA partnC // -order_dvdn dvdn_mulr.
rewrite -(@Gauss_dvdr _ #[x ^+ m]); last first.
by rewrite coprime_sym (pnat_coprime pi_xm) ?part_pnat.
apply: (@dvdn_trans #[x]); first by rewrite -{2}[#[x]](partnC pi) ?dvdn_mull.
by rewrite order_dvdn mulnC expgM expg_order.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
p_elt_exp
| |
mem_p_eltpi x G : pi.-group G -> x \in G -> pi.-elt x.
Proof. by move=> piG Gx; apply: pgroupS piG; rewrite cycle_subG. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
mem_p_elt
| |
p_eltM_normpi x y :
x \in 'N(<[y]>) -> pi.-elt x -> pi.-elt y -> pi.-elt (x * y).
Proof.
move=> nyx pi_x pi_y; apply: (@mem_p_elt pi _ (<[x]> <*> <[y]>)%G).
by rewrite /= norm_joinEl ?cycle_subG // pgroupM; apply/andP.
by rewrite groupM // mem_gen // inE cycle_id ?orbT.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
p_eltM_norm
| |
p_eltMpi x y : commute x y -> pi.-elt x -> pi.-elt y -> pi.-elt (x * y).
Proof.
move=> cxy; apply: p_eltM_norm; apply: (subsetP (cent_sub _)).
by rewrite cent_gen cent_set1; apply/cent1P.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
p_eltM
| |
p_elt1pi : pi.-elt (1 : gT).
Proof. by rewrite /p_elt order1. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
p_elt1
| |
p_eltVpi x : pi.-elt x^-1 = pi.-elt x.
Proof. by rewrite /p_elt orderV. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
p_eltV
| |
p_eltXpi x n : pi.-elt x -> pi.-elt (x ^+ n).
Proof. by rewrite -{1}[x]expg1 !p_elt_exp dvdn1 => /eqnP->. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
p_eltX
| |
p_eltJpi x y : pi.-elt (x ^ y) = pi.-elt x.
Proof. by congr pnat; rewrite orderJ. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
p_eltJ
| |
sub_p_eltpi1 pi2 x : {subset pi1 <= pi2} -> pi1.-elt x -> pi2.-elt x.
Proof. by move=> pi12; apply: sub_in_pnat => q _; apply: pi12. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
sub_p_elt
| |
eq_p_eltpi1 pi2 x : pi1 =i pi2 -> pi1.-elt x = pi2.-elt x.
Proof. by move=> pi12; apply: eq_pnat. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
eq_p_elt
| |
p_eltNKpi x : pi^'^'.-elt x = pi.-elt x.
Proof. exact: pnatNK. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
p_eltNK
| |
eq_consttpi1 pi2 x : pi1 =i pi2 -> x.`_pi1 = x.`_pi2.
Proof.
move=> pi12; congr (x ^+ (chinese _ _ 1 0)); apply: eq_partn => // a.
by congr (~~ _); apply: pi12.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
eq_constt
| |
consttNKpi x : x.`_pi^'^' = x.`_pi.
Proof. by rewrite /constt !partnNK. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
consttNK
| |
cycle_consttpi x : x.`_pi \in <[x]>.
Proof. exact: mem_cycle. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
cycle_constt
| |
consttVpi x : (x^-1).`_pi = (x.`_pi)^-1.
Proof. by rewrite /constt expgVn orderV. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
consttV
| |
constt1pi : 1.`_pi = 1 :> gT.
Proof. exact: expg1n. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
constt1
| |
consttJpi x y : (x ^ y).`_pi = x.`_pi ^ y.
Proof. by rewrite /constt orderJ conjXg. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
consttJ
| |
p_elt_consttpi x : pi.-elt x.`_pi.
Proof. by rewrite p_elt_exp /chinese addn0 mul1n dvdn_mulr. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
p_elt_constt
| |
consttCpi x : x.`_pi * x.`_pi^' = x.
Proof.
apply/eqP; rewrite -{3}[x]expg1 -expgD eq_expg_mod_order.
rewrite partnNK -{5 6}(@partnC pi #[x]) // /chinese !addn0.
by rewrite chinese_remainder ?chinese_modl ?chinese_modr ?coprime_partC ?eqxx.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
consttC
| |
p'_elt_consttpi x : pi^'.-elt (x * (x.`_pi)^-1).
Proof. by rewrite -{1}(consttC pi^' x) consttNK mulgK p_elt_constt. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
p'_elt_constt
| |
order_consttpi (x : gT) : #[x.`_pi] = (#[x]`_pi)%N.
Proof.
rewrite -{2}(consttC pi x) orderM; [|exact: commuteX2|]; last first.
by apply: (@pnat_coprime pi); apply: p_elt_constt.
by rewrite partnM // part_pnat_id ?part_p'nat ?muln1 //; apply: p_elt_constt.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
order_constt
| |
consttMpi x y : commute x y -> (x * y).`_pi = x.`_pi * y.`_pi.
Proof.
move=> cxy; pose m := #|<<[set x; y]>>|; have m_gt0: 0 < m := cardG_gt0 _.
pose k := chinese m`_pi m`_pi^' 1 0.
suffices kXpi z: z \in <<[set x; y]>> -> z.`_pi = z ^+ k.
by rewrite !kXpi ?expgMn // ?groupM ?mem_gen // !inE eqxx ?orbT.
move=> xyz; have{xyz} zm: #[z] %| m by rewrite cardSg ?cycle_subG.
apply/eqP; rewrite eq_expg_mod_order -{3 4}[#[z]](partnC pi) //.
rewrite chinese_remainder ?chinese_modl ?chinese_modr ?coprime_partC //.
rewrite -!(modn_dvdm k (partn_dvd _ m_gt0 zm)).
rewrite chinese_modl ?chinese_modr ?coprime_partC //.
by rewrite !modn_dvdm ?partn_dvd ?eqxx.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
consttM
| |
consttXpi x n : (x ^+ n).`_pi = x.`_pi ^+ n.
Proof.
elim: n => [|n IHn]; first exact: constt1.
by rewrite !expgS consttM ?IHn //; apply: commuteX.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
consttX
| |
constt1Ppi x : reflect (x.`_pi = 1) (pi^'.-elt x).
Proof.
rewrite -{2}[x]expg1 p_elt_exp -order_constt consttNK order_dvdn expg1.
exact: eqP.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
constt1P
| |
constt_p_eltpi x : pi.-elt x -> x.`_pi = x.
Proof.
by rewrite -p_eltNK -{3}(consttC pi x) => /constt1P->; rewrite mulg1.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
constt_p_elt
| |
sub_in_consttpi1 pi2 x :
{in \pi(#[x]), {subset pi1 <= pi2}} -> x.`_pi2.`_pi1 = x.`_pi1.
Proof.
move=> pi12; rewrite -{2}(consttC pi2 x) consttM; last exact: commuteX2.
rewrite (constt1P _ x.`_pi2^' _) ?mulg1 //.
apply: sub_in_pnat (p_elt_constt _ x) => p; rewrite order_constt => pi_p.
by apply/contra/pi12; rewrite -[#[x]](partnC pi2^') // primesM // pi_p.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
sub_in_constt
| |
prod_consttx : \prod_(0 <= p < #[x].+1) x.`_p = x.
Proof.
pose lp n := [pred p | p < n].
have: (lp #[x].+1).-elt x by apply/pnatP=> // p _; apply: dvdn_leq.
move/constt_p_elt=> def_x; symmetry; rewrite -{1}def_x {def_x}.
elim: _.+1 => [|p IHp].
by rewrite big_nil; apply/constt1P; apply/pgroupP.
rewrite big_nat_recr //= -{}IHp -(consttC (lp p) x.`__); congr (_ * _).
by rewrite sub_in_constt // => q _; apply: leqW.
set y := _.`__; rewrite -(consttC p y) (constt1P p^' _ _) ?mulg1.
by rewrite 2?sub_in_constt // => q _; move/eqnP->; rewrite !inE ?ltnn.
rewrite /p_elt pnatNK !order_constt -partnI.
apply: sub_in_pnat (part_pnat _ _) => q _.
by rewrite !inE ltnS -leqNgt -eqn_leq.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
prod_constt
| |
max_pgroupJpi M G x :
x \in G -> [max M | pi.-subgroup(G) M] ->
[max M :^ x of M | pi.-subgroup(G) M].
Proof.
move=> Gx /maxgroupP[piM maxM]; apply/maxgroupP.
split=> [|H piH]; first by rewrite psubgroupJ.
by rewrite -(conjsgKV x H) conjSg => /maxM/=-> //; rewrite psubgroupJ ?groupV.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
max_pgroupJ
| |
comm_sub_max_pgrouppi H M G :
[max M | pi.-subgroup(G) M] -> pi.-group H -> H \subset G ->
commute H M -> H \subset M.
Proof.
case/maxgroupP=> /andP[sMG piM] maxM piH sHG cHM.
rewrite -(maxM (H <*> M)%G) /= comm_joingE ?(mulG_subl, mulG_subr) //.
by rewrite /psubgroup pgroupM piM piH mul_subG.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
comm_sub_max_pgroup
| |
normal_sub_max_pgrouppi H M G :
[max M | pi.-subgroup(G) M] -> pi.-group H -> H <| G -> H \subset M.
Proof.
move=> maxM piH /andP[sHG nHG].
apply: comm_sub_max_pgroup piH sHG _ => //; apply: commute_sym; apply: normC.
by apply: subset_trans nHG; case/andP: (maxgroupp maxM).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
normal_sub_max_pgroup
| |
norm_sub_max_pgrouppi H M G :
[max M | pi.-subgroup(G) M] -> pi.-group H -> H \subset G ->
H \subset 'N(M) -> H \subset M.
Proof. by move=> maxM piH sHG /normC; apply: comm_sub_max_pgroup piH sHG. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
norm_sub_max_pgroup
| |
sub_pHallpi H G K :
pi.-Hall(G) H -> pi.-group K -> H \subset K -> K \subset G -> K :=: H.
Proof.
move=> hallH piK sHK sKG; apply/eqP; rewrite eq_sym eqEcard sHK.
by rewrite (card_Hall hallH) -(part_pnat_id piK) dvdn_leq ?partn_dvd ?cardSg.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
sub_pHall
| |
Hall_maxpi H G : pi.-Hall(G) H -> [max H | pi.-subgroup(G) H].
Proof.
move=> hallH; apply/maxgroupP; split=> [|K /andP[sKG piK] sHK].
by rewrite /psubgroup; case/and3P: hallH => ->.
exact: (sub_pHall hallH).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
Hall_max
| |
pHall_idpi H G : pi.-Hall(G) H -> pi.-group G -> H :=: G.
Proof.
by move=> hallH piG; rewrite (sub_pHall hallH piG) ?(pHall_sub hallH).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pHall_id
| |
psubgroup1pi G : pi.-subgroup(G) 1.
Proof. by rewrite /psubgroup sub1G pgroup1. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
psubgroup1
| |
Cauchyp G : prime p -> p %| #|G| -> {x | x \in G & #[x] = p}.
Proof.
move=> p_pr; have [n] := ubnP #|G|; elim: n G => // n IHn G /ltnSE-leGn pG.
pose xpG := [pred x in G | #[x] == p].
have [x /andP[Gx /eqP] | no_x] := pickP xpG; first by exists x.
have{pG n leGn IHn} pZ: p %| #|'C_G(G)|.
suffices /dvdn_addl <-: p %| #|G :\: 'C(G)| by rewrite cardsID.
have /acts_sum_card_orbit <-: [acts G, on G :\: 'C(G) | 'J].
by apply/actsP=> x Gx y; rewrite !inE -!mem_conjgV -centJ conjGid ?groupV.
elim/big_rec: _ => // _ _ /imsetP[x /setDP[Gx nCx] ->] /dvdn_addl->.
have ltCx: 'C_G[x] \proper G by rewrite properE subsetIl subsetIidl sub_cent1.
have /negP: ~ p %| #|'C_G[x]|.
case/(IHn _ (leq_trans (proper_card ltCx) leGn))=> y /setIP[Gy _] /eqP-oy.
by have /andP[] := no_x y.
by apply/implyP; rewrite -index_cent1 indexgI implyNb -Euclid_dvdM ?LagrangeI.
have [Q maxQ _]: {Q | [max Q | p^'.-subgroup('C_G(G)) Q] & 1%G \subset Q}.
by apply: maxgroup_exists; apply: psubgroup1.
case/andP: (maxgroupp maxQ) => sQC; rewrite /pgroup p'natE // => /negP[].
apply: dvdn_trans pZ (cardSg _); apply/subsetP=> x /setIP[Gx Cx].
rewrite -sub1set -gen_subG (normal_sub_max_pgroup maxQ) //; last first.
rewrite /normal subsetI !cycle_subG ?Gx ?cents_norm ?subIset ?andbT //=.
by rewrite centsC cycle_subG Cx.
rewrite /pgroup p'natE //= -[#|_|]/#[x]; apply/dvdnP=> [[m oxm]].
have m_gt0: 0 < m by apply: dvdn_gt0 (order_gt0 x) _; rewrite oxm dvdn_mulr.
case/idP: (no_x (x ^+ m)); rewrite /= groupX
...
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
Cauchy
| |
sub_normal_Hallpi G H K :
pi.-Hall(G) H -> H <| G -> K \subset G -> (K \subset H) = pi.-group K.
Proof.
move=> hallH nsHG sKG; apply/idP/idP=> [sKH | piK].
by rewrite (pgroupS sKH) ?(pHall_pgroup hallH).
apply: norm_sub_max_pgroup (Hall_max hallH) piK _ _ => //.
exact: subset_trans sKG (normal_norm nsHG).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
sub_normal_Hall
| |
mem_normal_Hallpi H G x :
pi.-Hall(G) H -> H <| G -> x \in G -> (x \in H) = pi.-elt x.
Proof. by rewrite -!cycle_subG; apply: sub_normal_Hall. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
mem_normal_Hall
| |
uniq_normal_Hallpi H G K :
pi.-Hall(G) H -> H <| G -> [max K | pi.-subgroup(G) K] -> K :=: H.
Proof.
move=> hallH nHG /maxgroupP[/andP[sKG piK] /(_ H) -> //].
exact: (maxgroupp (Hall_max hallH)).
by rewrite (sub_normal_Hall hallH).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
uniq_normal_Hall
| |
normal_max_pgroup_HallG H :
[max H | pi.-subgroup(G) H] -> H <| G -> pi.-Hall(G) H.
Proof.
case/maxgroupP=> /andP[sHG piH] maxH nsHG; have [_ nHG] := andP nsHG.
rewrite /pHall sHG piH; apply/pnatP=> // p p_pr.
rewrite inE /= -pnatE // -card_quotient //.
case/Cauchy=> //= Hx; rewrite -sub1set -gen_subG -/<[Hx]> /order.
case/inv_quotientS=> //= K -> sHK sKG {Hx}.
rewrite card_quotient ?(subset_trans sKG) // => iKH; apply/negP=> pi_p.
rewrite -iKH -divgS // (maxH K) ?divnn ?cardG_gt0 // in p_pr.
by rewrite /psubgroup sKG /pgroup -(Lagrange sHK) mulnC pnatM iKH pi_p.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
normal_max_pgroup_Hall
| |
setI_normal_HallG H K :
H <| G -> pi.-Hall(G) H -> K \subset G -> pi.-Hall(K) (H :&: K).
Proof.
move=> nsHG hallH sKG; apply: normal_max_pgroup_Hall; last first.
by rewrite /= setIC (normalGI sKG nsHG).
apply/maxgroupP; split=> [|M /andP[sMK piM] sHK_M].
by rewrite /psubgroup subsetIr (pgroupS (subsetIl _ _) (pHall_pgroup hallH)).
apply/eqP; rewrite eqEsubset sHK_M subsetI sMK !andbT.
by rewrite (sub_normal_Hall hallH) // (subset_trans sMK).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
setI_normal_Hall
| |
morphim_pgrouppi G : pi.-group G -> pi.-group (f @* G).
Proof. by apply: pnat_dvd; apply: dvdn_morphim. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
morphim_pgroup
| |
morphim_oddG : odd #|G| -> odd #|f @* G|.
Proof. by rewrite !odd_2'nat; apply: morphim_pgroup. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
morphim_odd
| |
pmorphim_pgrouppi G :
pi.-group ('ker f) -> G \subset D -> pi.-group (f @* G) = pi.-group G.
Proof.
move=> piker sGD; apply/idP/idP=> [pifG|]; last exact: morphim_pgroup.
apply: (@pgroupS _ _ (f @*^-1 (f @* G))); first by rewrite -sub_morphim_pre.
by rewrite /pgroup card_morphpre ?morphimS // pnatM; apply/andP.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pmorphim_pgroup
| |
morphim_p_indexpi G H :
H \subset D -> pi.-nat #|G : H| -> pi.-nat #|f @* G : f @* H|.
Proof.
by move=> sHD; apply: pnat_dvd; rewrite index_morphim ?subIset // sHD orbT.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
morphim_p_index
| |
morphim_pHallpi G H :
H \subset D -> pi.-Hall(G) H -> pi.-Hall(f @* G) (f @* H).
Proof.
move=> sHD /and3P[sHG piH pi'GH].
by rewrite /pHall morphimS // morphim_pgroup // morphim_p_index.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
morphim_pHall
| |
pmorphim_pHallpi G H :
G \subset D -> H \subset D -> pi.-subgroup(H :&: G) ('ker f) ->
pi.-Hall(f @* G) (f @* H) = pi.-Hall(G) H.
Proof.
move=> sGD sHD /andP[/subsetIP[sKH sKG] piK]; rewrite !pHallE morphimSGK //.
apply: andb_id2l => sHG; rewrite -(Lagrange sKH) -(Lagrange sKG) partnM //.
by rewrite (part_pnat_id piK) !card_morphim !(setIidPr _) // eqn_pmul2l.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
pmorphim_pHall
| |
morphim_HallG H : H \subset D -> Hall G H -> Hall (f @* G) (f @* H).
Proof.
by move=> sHD /HallP[pi piH]; apply: (@pHall_Hall _ pi); apply: morphim_pHall.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
morphim_Hall
|
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