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 |
|---|---|---|---|---|---|---|
charsimplePG :
reflect (G :!=: 1 /\ forall K, K :!=: 1 -> K \char G -> K :=: G)
(charsimple G).
Proof.
apply: (iffP mingroupP); rewrite char_refl andbT => -[ntG simG].
by split=> // K ntK chK; apply: simG; rewrite ?ntK // char_sub.
by split=> // K /andP[ntK chK] _; apply: simG.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | charsimpleP | |
Fitting_normalG : 'F(G) <| G.
Proof.
rewrite -['F(G)](bigdprodWY (erefl 'F(G))).
elim/big_rec: _ => [|p H _ nsHG]; first by rewrite gen0 normal1.
by rewrite -[<<_>>]joing_idr normalY ?pcore_normal.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Fitting_normal | |
Fitting_subG : 'F(G) \subset G.
Proof. by rewrite normal_sub ?Fitting_normal. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Fitting_sub | |
Fitting_nilG : nilpotent 'F(G).
Proof.
apply: (bigdprod_nil (erefl 'F(G))) => p _.
exact: pgroup_nil (pcore_pgroup p G).
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Fitting_nil | |
Fitting_maxG H : H <| G -> nilpotent H -> H \subset 'F(G).
Proof.
move=> nsHG nilH; rewrite -(Sylow_gen H) gen_subG.
apply/bigcupsP=> P /SylowP[p _ sylP].
case Gp: (p \in \pi(G)); last first.
rewrite card1_trivg ?sub1G // (card_Hall sylP).
rewrite part_p'nat // (pnat_dvd (cardSg (normal_sub nsHG))) //.
by rewrite... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Fitting_max | |
pcore_Fittingpi G : 'O_pi('F(G)) \subset 'O_pi(G).
Proof. by rewrite pcore_max ?pcore_pgroup ?gFnormal_trans ?Fitting_normal. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | pcore_Fitting | |
p_core_Fittingp G : 'O_p('F(G)) = 'O_p(G).
Proof.
apply/eqP; rewrite eqEsubset pcore_Fitting pcore_max ?pcore_pgroup //.
apply: normalS (normal_sub (Fitting_normal _)) (pcore_normal _ _).
exact: Fitting_max (pcore_normal _ _) (pgroup_nil (pcore_pgroup _ _)).
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | p_core_Fitting | |
nilpotent_FittingG : nilpotent G -> 'F(G) = G.
Proof.
by move=> nilG; apply/eqP; rewrite eqEsubset Fitting_sub Fitting_max.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | nilpotent_Fitting | |
Fitting_eq_pcorep G : 'O_p^'(G) = 1 -> 'F(G) = 'O_p(G).
Proof.
move=> p'G1; have /dprodP[_ /= <- _ _] := nilpotent_pcoreC p (Fitting_nil G).
by rewrite p_core_Fitting ['O_p^'(_)](trivgP _) ?mulg1 // -p'G1 pcore_Fitting.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Fitting_eq_pcore | |
FittingEgenG :
'F(G) = <<\bigcup_(p < #|G|.+1 | (p : nat) \in \pi(G)) 'O_p(G)>>.
Proof.
apply/eqP; rewrite eqEsubset gen_subG /=.
rewrite -{1}(bigdprodWY (erefl 'F(G))) (big_nth 0) big_mkord genS.
by apply/bigcupsP=> p _; rewrite -p_core_Fitting pcore_sub.
apply/bigcupsP=> [[i /= lti]] _; set p := nth _ _ i.
have p... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | FittingEgen | |
morphim_Fitting: GFunctor.pcontinuous (@Fitting).
Proof.
move=> gT rT G D f; apply: Fitting_max.
by rewrite morphim_normal ?Fitting_normal.
by rewrite morphim_nil ?Fitting_nil.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | morphim_Fitting | |
FittingSgT (G H : {group gT}) : H \subset G -> H :&: 'F(G) \subset 'F(H).
Proof.
move=> sHG; rewrite -{2}(setIidPl sHG).
do 2!rewrite -(morphim_idm (subsetIl H _)) morphimIdom; apply: morphim_Fitting.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | FittingS | |
FittingJgT (G : {group gT}) x : 'F(G :^ x) = 'F(G) :^ x.
Proof.
rewrite !FittingEgen -genJ /= cardJg; symmetry; congr <<_>>.
rewrite (big_morph (conjugate^~ x) (fun A B => conjUg A B x) (imset0 _)).
by apply: eq_bigr => p _; rewrite pcoreJ.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | FittingJ | |
Fitting_igFun:= [igFun by Fitting_sub & morphim_Fitting]. | Canonical | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Fitting_igFun | |
Fitting_gFun:= [gFun by morphim_Fitting]. | Canonical | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Fitting_gFun | |
Fitting_pgFun:= [pgFun by morphim_Fitting]. | Canonical | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Fitting_pgFun | |
Fitting_char: 'F(G) \char G. Proof. exact: gFchar. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Fitting_char | |
injm_Fitting: 'injm f -> G \subset D -> f @* 'F(G) = 'F(f @* G).
Proof. exact: injmF. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | injm_Fitting | |
isog_Fitting(H : {group rT}) : G \isog H -> 'F(G) \isog 'F(H).
Proof. exact: gFisog. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | isog_Fitting | |
minnormal_charsimpleG H : minnormal H G -> charsimple H.
Proof.
case/mingroupP=> /andP[ntH nHG] minH.
apply/charsimpleP; split=> // K ntK chK.
by apply: minH; rewrite ?ntK (char_sub chK, char_norm_trans chK).
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | minnormal_charsimple | |
maxnormal_charsimpleG H L :
G <| L -> maxnormal H G L -> charsimple (G / H).
Proof.
case/andP=> sGL nGL /maxgroupP[/andP[/andP[sHG not_sGH] nHL] maxH].
have nHG: G \subset 'N(H) := subset_trans sGL nHL.
apply/charsimpleP; rewrite -subG1 quotient_sub1 //; split=> // HK ntHK chHK.
case/(inv_quotientN _): (char_normal c... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | maxnormal_charsimple | |
abelem_split_dprodrT p (A B : {group rT}) :
p.-abelem A -> B \subset A -> exists C : {group rT}, B \x C = A.
Proof.
move=> abelA sBA; have [_ cAA _]:= and3P abelA.
case/splitsP: (abelem_splits abelA sBA) => C /complP[tiBC defA].
by exists C; rewrite dprodE // (centSS _ sBA cAA) // -defA mulG_subr.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | abelem_split_dprod | |
p_abelem_split1rT p (A : {group rT}) x :
p.-abelem A -> x \in A ->
exists B : {group rT}, [/\ B \subset A, #|B| = #|A| %/ #[x] & <[x]> \x B = A].
Proof.
move=> abelA Ax; have sxA: <[x]> \subset A by rewrite cycle_subG.
have [B defA] := abelem_split_dprod abelA sxA.
have [_ defxB _ ti_xB] := dprodP defA.
have sBA... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | p_abelem_split1 | |
abelem_charsimplep G : p.-abelem G -> G :!=: 1 -> charsimple G.
Proof.
move=> abelG ntG; apply/charsimpleP; split=> // K ntK /charP[sKG chK].
case/eqVproper: sKG => // /properP[sKG [x Gx notKx]].
have ox := abelem_order_p abelG Gx (group1_contra notKx).
have [A [sAG oA defA]] := p_abelem_split1 abelG Gx.
case/trivgPn: ... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | abelem_charsimple | |
charsimple_dprodG : charsimple G ->
exists H : {group gT}, [/\ H \subset G, simple H
& exists2 I : {set {perm gT}}, I \subset Aut G
& \big[dprod/1]_(f in I) f @: H = G].
Proof.
case/charsimpleP=> ntG simG.
have [H minH sHG]: {H : {group gT} | minnormal H G & H \subset... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | charsimple_dprod | |
simple_sol_primeG : solvable G -> simple G -> prime #|G|.
Proof.
move=> solG /simpleP[ntG simG].
have{solG} cGG: abelian G.
apply/commG1P; case/simG: (der_normal 1 G) => // /eqP/idPn[].
by rewrite proper_neq // (sol_der1_proper solG).
case: (trivgVpdiv G) ntG => [-> | [p p_pr]]; first by rewrite eqxx.
case/Cauchy=>... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | simple_sol_prime | |
charsimple_solvableG : charsimple G -> solvable G -> is_abelem G.
Proof.
case/charsimple_dprod=> H [sHG simH [I Aut_I defG]] solG.
have p_pr: prime #|H| by apply: simple_sol_prime (solvableS sHG solG) simH.
set p := #|H| in p_pr; apply/is_abelemP; exists p => //.
elim/big_rec: _ (G) defG => [_ <-|f B If IH_B M defM]; f... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | charsimple_solvable | |
minnormal_solvableL G H :
minnormal H L -> H \subset G -> solvable G ->
[/\ L \subset 'N(H), H :!=: 1 & is_abelem H].
Proof.
move=> minH sHG solG; have /andP[ntH nHL] := mingroupp minH.
split=> //; apply: (charsimple_solvable (minnormal_charsimple minH)).
exact: solvableS solG.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | minnormal_solvable | |
solvable_norm_abelemL G :
solvable G -> G <| L -> G :!=: 1 ->
exists H : {group gT}, [/\ H \subset G, H <| L, H :!=: 1 & is_abelem H].
Proof.
move=> solG /andP[sGL nGL] ntG.
have [H minH sHG]: {H : {group gT} | minnormal H L & H \subset G}.
by apply: mingroup_exists; rewrite ntG.
have [nHL ntH abH] := minnormal... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | solvable_norm_abelem | |
trivg_FittingG : solvable G -> ('F(G) == 1) = (G :==: 1).
Proof.
move=> solG; apply/idP/idP=> [F1 | /eqP->]; last by rewrite gF1.
apply/idPn=> /(solvable_norm_abelem solG (normal_refl _))[M [_ nsMG ntM]].
case/is_abelemP=> p _ /and3P[pM _ _]; case/negP: ntM.
by rewrite -subG1 -(eqP F1) Fitting_max ?(pgroup_nil pM).
Qed... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | trivg_Fitting | |
Fitting_pcorepi G : 'F('O_pi(G)) = 'O_pi('F(G)).
Proof.
apply/eqP; rewrite eqEsubset.
rewrite (subset_trans _ (pcoreS _ (Fitting_sub _))); last first.
by rewrite subsetI Fitting_sub Fitting_max ?Fitting_nil ?gFnormal_trans.
rewrite (subset_trans _ (FittingS (pcore_sub _ _))) // subsetI pcore_sub.
by rewrite pcore_max... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Fitting_pcore | |
index_maxnormal_sol_prime(H : {group gT}) :
solvable G -> maxnormal H G G -> prime #|G : H|.
Proof.
move=> solG maxH; have nsHG := maxnormal_normal maxH.
rewrite -card_quotient ?normal_norm // simple_sol_prime ?quotient_sol //.
by rewrite quotient_simple.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | index_maxnormal_sol_prime | |
sol_prime_factor_exists:
solvable G -> G :!=: 1 -> {H : {group gT} | H <| G & prime #|G : H| }.
Proof.
move=> solG /ex_maxnormal_ntrivg[H maxH].
by exists H; [apply: maxnormal_normal | apply: index_maxnormal_sol_prime].
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | sol_prime_factor_exists | |
center_special_abelem: p.-group G -> special G -> p.-abelem 'Z(G).
Proof.
move=> pG [defPhi defG'].
have [-> | ntG] := eqsVneq G 1; first by rewrite center1 abelem1.
have [p_pr _ _] := pgroup_pdiv pG ntG.
have fM: {in 'Z(G) &, {morph natexp^~ p : x y / x * y}}.
by move=> x y /setIP[_ /centP cxG] /setIP[/cxG cxy _]; a... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | center_special_abelem | |
exponent_special: p.-group G -> special G -> exponent G %| p ^ 2.
Proof.
move=> pG spG; have [defPhi _] := spG.
have /and3P[_ _ expZ] := center_special_abelem pG spG.
apply/exponentP=> x Gx; rewrite expgM (exponentP expZ) // -defPhi.
by rewrite (Phi_joing pG) mem_gen // inE orbC (Mho_p_elt 1) ?(mem_p_elt pG).
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | exponent_special | |
abelian_charsimple_special:
p.-group G -> coprime #|G| #|A| -> [~: G, A] = G ->
\bigcup_(H : {group gT} | (H \char G) && abelian H) H \subset 'C(A) ->
special G /\ 'C_G(A) = 'Z(G).
Proof.
move=> pG coGA defG /bigcupsP cChaA.
have cZA: 'Z(G) \subset 'C_G(A).
by rewrite subsetI center_sub cChaA // center_char... | Theorem | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | abelian_charsimple_special | |
extraspecial_prime: prime p.
Proof.
by case: esS => _ /prime_gt1; rewrite cardG_gt1; case/(pgroup_pdiv pZ).
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | extraspecial_prime | |
card_center_extraspecial: #|'Z(S)| = p.
Proof. by apply/eqP; apply: (pgroupP pZ); case: esS. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | card_center_extraspecial | |
min_card_extraspecial: #|S| >= p ^ 3.
Proof.
have p_gt1 := prime_gt1 extraspecial_prime.
rewrite leqNgt (card_pgroup pS) ltn_exp2l // ltnS.
case: esS => [[_ defS']]; apply: contraL => /(p2group_abelian pS)/derG1P S'1.
by rewrite -defS' S'1 cards1.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | min_card_extraspecial | |
card_p3group_extraspecialE :
prime p -> #|E| = (p ^ 3)%N -> #|'Z(E)| = p -> extraspecial E.
Proof.
move=> p_pr oEp3 oZp; have p_gt0 := prime_gt0 p_pr.
have pE: p.-group E by rewrite /pgroup oEp3 pnatX pnat_id.
have pEq: p.-group (E / 'Z(E))%g by rewrite quotient_pgroup.
have /andP[sZE nZE] := center_normal E.
have oE... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | card_p3group_extraspecial | |
p3group_extraspecialG :
p.-group G -> ~~ abelian G -> logn p #|G| <= 3 -> extraspecial G.
Proof.
move=> pG not_cGG; have /andP[sZG nZG] := center_normal G.
have ntG: G :!=: 1 by apply: contraNneq not_cGG => ->; apply: abelian1.
have ntZ: 'Z(G) != 1 by rewrite (center_nil_eq1 (pgroup_nil pG)).
have [p_pr _ [n oG]] := ... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | p3group_extraspecial | |
extraspecial_nonabelianG : extraspecial G -> ~~ abelian G.
Proof.
case=> [[_ defG'] oZ]; rewrite /abelian (sameP commG1P eqP).
by rewrite -derg1 defG' -cardG_gt1 prime_gt1.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | extraspecial_nonabelian | |
exponent_2extraspecialG : 2.-group G -> extraspecial G -> exponent G = 4.
Proof.
move=> p2G esG; have [spG _] := esG.
case/dvdn_pfactor: (exponent_special p2G spG) => // k.
rewrite leq_eqVlt ltnS => /predU1P[-> // | lek1] expG.
case/negP: (extraspecial_nonabelian esG).
by rewrite (@abelem_abelian _ 2) ?exponent2_abelem... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | exponent_2extraspecial | |
injm_specialD G (f : {morphism D >-> rT}) :
'injm f -> G \subset D -> special G -> special (f @* G).
Proof.
move=> injf sGD [defPhiG defG'].
by rewrite /special -morphim_der // -injm_Phi // defPhiG defG' injm_center.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | injm_special | |
injm_extraspecialD G (f : {morphism D >-> rT}) :
'injm f -> G \subset D -> extraspecial G -> extraspecial (f @* G).
Proof.
move=> injf sGD [spG ZG_pr]; split; first exact: injm_special spG.
by rewrite -injm_center // card_injm // subIset ?sGD.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | injm_extraspecial | |
isog_specialG (R : {group rT}) :
G \isog R -> special G -> special R.
Proof. by case/isogP=> f injf <-; apply: injm_special. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | isog_special | |
isog_extraspecialG (R : {group rT}) :
G \isog R -> extraspecial G -> extraspecial R.
Proof. by case/isogP=> f injf <-; apply: injm_extraspecial. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | isog_extraspecial | |
cprod_extraspecialG H K :
p.-group G -> H \* K = G -> H :&: K = 'Z(H) ->
extraspecial H -> extraspecial K -> extraspecial G.
Proof.
move=> pG defG ziHK [[PhiH defH'] ZH_pr] [[PhiK defK'] ZK_pr].
have [_ defHK cHK]:= cprodP defG.
have sZHK: 'Z(H) \subset 'Z(K).
by rewrite subsetI -{1}ziHK subsetIr subIset // cen... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | cprod_extraspecial | |
cent1_extraspecial_maximalx :
x \in G -> x \notin 'Z(G) -> maximal 'C_G[x] G.
Proof.
move=> Gx notZx; pose f y := [~ x, y]; have [[_ defG'] prZ] := esG.
have{defG'} fZ y: y \in G -> f y \in 'Z(G).
by move=> Gy; rewrite -defG' mem_commg.
have fM: {in G &, {morph f : y z / y * z}}%g.
move=> y z Gy Gz; rewrite {1}/f... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | cent1_extraspecial_maximal | |
subcent1_extraspecial_maximalU x :
U \subset G -> x \in G :\: 'C(U) -> maximal 'C_U[x] U.
Proof.
move=> sUG /setDP[Gx not_cUx]; apply/maxgroupP; split=> [|H ltHU sCxH].
by rewrite /proper subsetIl subsetI subxx sub_cent1.
case/andP: ltHU => sHU not_sHU; have sHG := subset_trans sHU sUG.
apply/eqP; rewrite eqEsubset... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | subcent1_extraspecial_maximal | |
card_subcent_extraspecialU :
U \subset G -> #|'C_G(U)| = (#|'Z(G) :&: U| * #|G : U|)%N.
Proof.
move=> sUG; rewrite setIAC (setIidPr sUG).
have [m leUm] := ubnP #|U|; elim: m => // m IHm in U leUm sUG *.
have [cUG | not_cUG]:= orP (orbN (G \subset 'C(U))).
by rewrite !(setIidPl _) ?Lagrange // centsC.
have{not_cUG} ... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | card_subcent_extraspecial | |
split1_extraspecialx :
x \in G :\: 'Z(G) ->
{E : {group gT} & {R : {group gT} |
[/\ #|E| = (p ^ 3)%N /\ #|R| = #|G| %/ p ^ 2,
E \* R = G /\ E :&: R = 'Z(E),
'Z(E) = 'Z(G) /\ 'Z(R) = 'Z(G),
extraspecial E /\ x \in E
& if abelian R then R :=: 'Z(G) else extraspecial R]}}.
Proof.
ca... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | split1_extraspecial | |
pmaxElem_extraspecial: 'E*_p(G) = 'E_p^('r_p(G))(G).
Proof.
have sZmax: {in 'E*_p(G), forall E, 'Z(G) \subset E}.
move=> E maxE; have defE := pmaxElem_LdivP p_pr maxE.
have abelZ: p.-abelem 'Z(G) by rewrite prime_abelem ?oZ.
rewrite -(Ohm1_id abelZ) (OhmE 1 (abelem_pgroup abelZ)) gen_subG -defE.
by rewrite setS... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | pmaxElem_extraspecial | |
critical_extraspecialR S :
p.-group R -> S \subset R -> extraspecial S -> [~: S, R] \subset S^`(1) ->
S \* 'C_R(S) = R.
Proof.
move=> pR sSR esS sSR_S'; have [[defPhi defS'] _] := esS.
have [pS [sPS nPS]] := (pgroupS sSR pR, andP (Phi_normal S : 'Phi(S) <| S)).
have{esS} oZS: #|'Z(S)| = p := card_center_extraspec... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | critical_extraspecial | |
extraspecial_structureS : p.-group S -> extraspecial S ->
{Es | all (fun E => (#|E| == p ^ 3)%N && ('Z(E) == 'Z(S))) Es
& \big[cprod/1%g]_(E <- Es) E \* 'Z(S) = S}.
Proof.
have [m] := ubnP #|S|; elim: m S => // m IHm S leSm pS esS.
have [x Z'x]: {x | x \in S :\: 'Z(S)}.
apply/sigW/set0Pn; rewrite -subset0 sub... | Theorem | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | extraspecial_structure | |
card_extraspecial: {n | n > 0 & #|S| = (p ^ n.*2.+1)%N}.
Proof.
set T := S; exists (logn p #|T|)./2.
rewrite half_gt0 ltnW // -(leq_exp2l _ _ (prime_gt1 p_pr)) -card_pgroup //.
exact: min_card_extraspecial.
have [Es] := extraspecial_structure pS esS; rewrite -[in RHS]/T.
elim: Es T => [_ _ <-| E s IHs T] /=.
by r... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | card_extraspecial | |
Aut_extraspecial_full: Aut_in (Aut S) 'Z(S) \isog Aut 'Z(S).
Proof.
have [p_gt1 p_gt0] := (prime_gt1 p_pr, prime_gt0 p_pr).
have [Es] := extraspecial_structure pS esS.
elim: Es S oZ => [T _ _ <-| E s IHs T oZT] /=.
rewrite big_nil cprod1g (center_idP (center_abelian T)).
by apply/Aut_sub_fullP=> // g injg gZ; exist... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Aut_extraspecial_full | |
center_aut_extraspecialk : coprime k p ->
exists2 f, f \in Aut S & forall z, z \in 'Z(S) -> f z = (z ^+ k)%g.
Proof.
have /cyclicP[z defZ]: cyclic 'Z(S) by rewrite prime_cyclic ?oZ.
have oz: #[z] = p by rewrite orderE -defZ.
rewrite coprime_sym -unitZpE ?prime_gt1 // -oz => u_k.
pose g := Zp_unitm (FinRing.unit 'Z_#[... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | center_aut_extraspecial | |
SCN_PA : reflect (A <| G /\ 'C_G(A) = A) (A \in 'SCN(G)).
Proof. by apply: (iffP setIdP) => [] [->]; move/eqP. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | SCN_P | |
SCN_abelianA : A \in 'SCN(G) -> abelian A.
Proof. by case/SCN_P=> _ defA; rewrite /abelian -{1}defA subsetIr. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | SCN_abelian | |
exponent_Ohm1_class2H :
odd p -> p.-group H -> nil_class H <= 2 -> exponent 'Ohm_1(H) %| p.
Proof.
move=> odd_p pH; rewrite nil_class2 => sH'Z; apply/exponentP=> x /=.
rewrite (OhmE 1 pH) expn1 gen_set_id => {x} [/LdivP[] //|].
apply/group_setP; split=> [|x y]; first by rewrite !inE group1 expg1n //=.
case/LdivP=> Hx... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | exponent_Ohm1_class2 | |
SCN_maxA : A \in 'SCN(G) -> [max A | A <| G & abelian A].
Proof.
case/SCN_P => nAG scA; apply/maxgroupP; split=> [|H].
by rewrite nAG /abelian -{1}scA subsetIr.
do 2![case/andP]=> sHG _ abelH sAH; apply/eqP.
by rewrite eqEsubset sAH -scA subsetI sHG centsC (subset_trans sAH).
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | SCN_max | |
max_SCNA :
p.-group G -> [max A | A <| G & abelian A] -> A \in 'SCN(G).
Proof.
move/pgroup_nil=> nilG; rewrite /abelian.
case/maxgroupP=> /andP[nsAG abelA] maxA; have [sAG nAG] := andP nsAG.
rewrite inE nsAG eqEsubset /= andbC subsetI abelA normal_sub //=.
rewrite -quotient_sub1; last by rewrite subIset 1?normal_norm... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | max_SCN | |
der1_stab_Ohm1_SCN_series: ('C(Z) :&: 'C_G(A / Z | 'Q))^`(1) \subset A.
Proof.
case/SCN_P: SCN_A => /andP[sAG nAG] {4} <-.
rewrite subsetI {1}setICA comm_subG ?subsetIl //= gen_subG.
apply/subsetP=> w /imset2P[u v].
rewrite /= -groupV -(groupV _ v) /= astabQR //= -/Z !inE (groupV 'C(Z)).
case/and4P=> cZu _ _ sRuZ /and4... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | der1_stab_Ohm1_SCN_series | |
Ohm1_stab_Ohm1_SCN_series:
odd p -> p.-group G -> 'Ohm_1('C_G(Z)) \subset 'C_G(A / Z | 'Q).
Proof.
have [-> | ntG] := eqsVneq G 1; first by rewrite !(setIidPl (sub1G _)) Ohm1.
move=> p_odd pG; have{ntG} [p_pr _ _] := pgroup_pdiv pG ntG.
case/SCN_P: SCN_A => /andP[sAG nAG] _; have pA := pgroupS sAG pG.
have pCGZ : p.-... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Ohm1_stab_Ohm1_SCN_series | |
Ohm1_cent_max_normal_abelemZ :
odd p -> p.-group G -> [max Z | Z <| G & p.-abelem Z] -> 'Ohm_1('C_G(Z)) = Z.
Proof.
move=> p_odd pG; set X := 'Ohm_1('C_G(Z)).
case/maxgroupP=> /andP[nsZG abelZ] maxZ.
have [sZG nZG] := andP nsZG; have [_ cZZ expZp] := and3P abelZ.
have{nZG} nsXG: X <| G by rewrite gFnormal_trans ?norm... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Ohm1_cent_max_normal_abelem | |
critical_class2H : critical H G -> nil_class H <= 2.
Proof.
case=> [chH _ sRZ _].
by rewrite nil_class2 (subset_trans _ sRZ) ?commSg // char_sub.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | critical_class2 | |
Thompson_critical: p.-group G -> {K : {group gT} | critical K G}.
Proof.
move=> pG; pose qcr A := (A \char G) && ('Phi(A) :|: [~: G, A] \subset 'Z(A)).
have [|K]:= @maxgroup_exists _ qcr 1 _.
by rewrite /qcr char1 center1 commG1 subUset Phi_sub subxx.
case/maxgroupP; rewrite {}/qcr subUset => /and3P[chK sPhiZ sRZ] ma... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | Thompson_critical | |
critical_p_stab_AutH :
critical H G -> p.-group G -> p.-group 'C(H | [Aut G]).
Proof.
move=> [chH sPhiZ sRZ eqCZ] pG; have sHG := char_sub chH.
pose G' := (sdpair1 [Aut G] @* G)%G; pose H' := (sdpair1 [Aut G] @* H)%G.
apply/pgroupP=> q pr_q; case/Cauchy=> //= f cHF; move: (cHF); rewrite astab_ract.
case/setIP=> Af cH... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype finfun bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action commutator gproduct gfunctor ssralg... | solvable/maximal.v | critical_p_stab_Aut | |
lower_central_at:= iter n.-1 (fun B => [~: B, A]) A. | Definition | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | lower_central_at | |
upper_central_at:= iter n (fun B => coset B @*^-1 'Z(A / B)) 1. | Definition | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | upper_central_at | |
nilpotent:=
[forall (G : {group gT} | G \subset A :&: [~: G, A]), G :==: 1]. | Definition | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | nilpotent | |
nil_class:= index 1 (mkseq (fun n => 'L_n.+1(A)) #|A|). | Definition | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | nil_class | |
solvable:=
[forall (G : {group gT} | G \subset A :&: [~: G, G]), G :==: 1]. | Definition | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | solvable | |
nilpotent1: nilpotent [1 gT].
Proof. by apply/forall_inP=> H; rewrite commG1 setIid -subG1. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | nilpotent1 | |
nilpotentSA B : B \subset A -> nilpotent A -> nilpotent B.
Proof.
move=> sBA nilA; apply/forall_inP=> H sHR.
have:= forallP nilA H; rewrite (subset_trans sHR) //.
by apply: subset_trans (setIS _ _) (setSI _ _); rewrite ?commgS.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | nilpotentS | |
nil_comm_properlG H A :
nilpotent G -> H \subset G -> H :!=: 1 -> A \subset 'N_G(H) ->
[~: H, A] \proper H.
Proof.
move=> nilG sHG ntH; rewrite subsetI properE; case/andP=> sAG nHA.
rewrite (subset_trans (commgS H (subset_gen A))) ?commg_subl ?gen_subG //.
apply: contra ntH => sHR; have:= forallP nilG H; rewrite ... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | nil_comm_properl | |
nil_comm_properrG A H :
nilpotent G -> H \subset G -> H :!=: 1 -> A \subset 'N_G(H) ->
[~: A, H] \proper H.
Proof. by rewrite commGC; apply: nil_comm_properl. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | nil_comm_properr | |
centrals_nil(s : seq {group gT}) G :
G.-central.-series 1%G s -> last 1%G s = G -> nilpotent G.
Proof.
move=> cGs defG; apply/forall_inP=> H /subsetIP[sHG sHR].
move: sHG; rewrite -{}defG -subG1 -[1]/(gval 1%G).
elim: s 1%G cGs => //= L s IHs K /andP[/and3P[sRK sKL sLG] /IHs sHL] sHs.
exact: subset_trans sHR (subset_... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | centrals_nil | |
lcn0A : 'L_0(A) = A. Proof. by []. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | lcn0 | |
lcn1A : 'L_1(A) = A. Proof. by []. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | lcn1 | |
lcnSnn A : 'L_n.+2(A) = [~: 'L_n.+1(A), A]. Proof. by []. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | lcnSn | |
lcnSnSn G : [~: 'L_n(G), G] \subset 'L_n.+1(G).
Proof. by case: n => //; apply: der1_subG. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | lcnSnS | |
lcnEn A : 'L_n.+1(A) = iter n (fun B => [~: B, A]) A.
Proof. by []. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | lcnE | |
lcn2A : 'L_2(A) = A^`(1). Proof. by []. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | lcn2 | |
lcn_group_setn G : group_set 'L_n(G).
Proof. by case: n => [|[|n]]; apply: groupP. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | lcn_group_set | |
lower_central_at_groupn G := Group (lcn_group_set n G). | Canonical | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | lower_central_at_group | |
lcn_charn G : 'L_n(G) \char G.
Proof. by case: n; last elim=> [|n IHn]; rewrite ?char_refl ?lcnSn ?charR. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | lcn_char | |
lcn_normaln G : 'L_n(G) <| G.
Proof. exact/char_normal/lcn_char. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | lcn_normal | |
lcn_subn G : 'L_n(G) \subset G.
Proof. exact/char_sub/lcn_char. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | lcn_sub | |
lcn_normn G : G \subset 'N('L_n(G)).
Proof. exact/char_norm/lcn_char. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | lcn_norm | |
lcn_subSn G : 'L_n.+1(G) \subset 'L_n(G).
Proof.
case: n => // n; rewrite lcnSn commGC commg_subr.
by case/andP: (lcn_normal n.+1 G).
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | lcn_subS | |
lcn_normalSn G : 'L_n.+1(G) <| 'L_n(G).
Proof. by apply: normalS (lcn_normal _ _); rewrite (lcn_subS, lcn_sub). Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | lcn_normalS | |
lcn_centraln G : 'L_n(G) / 'L_n.+1(G) \subset 'Z(G / 'L_n.+1(G)).
Proof.
case: n => [|n]; first by rewrite trivg_quotient sub1G.
by rewrite subsetI quotientS ?lcn_sub ?quotient_cents2r.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | lcn_central | |
lcn_sub_leqm n G : n <= m -> 'L_m(G) \subset 'L_n(G).
Proof.
by move/subnK <-; elim: {m}(m - n) => // m; apply: subset_trans (lcn_subS _ _).
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | lcn_sub_leq | |
lcnSn A B : A \subset B -> 'L_n(A) \subset 'L_n(B).
Proof.
by case: n => // n sAB; elim: n => // n IHn; rewrite !lcnSn genS ?imset2S.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | lcnS | |
lcn_cprodn A B G : A \* B = G -> 'L_n(A) \* 'L_n(B) = 'L_n(G).
Proof.
case: n => // n /cprodP[[H K -> ->{A B}] defG cHK].
have sL := subset_trans (lcn_sub _ _); rewrite cprodE ?(centSS _ _ cHK) ?sL //.
symmetry; elim: n => // n; rewrite lcnSn => ->; rewrite commMG /=; last first.
by apply: subset_trans (commg_normr _... | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | lcn_cprod | |
lcn_dprodn A B G : A \x B = G -> 'L_n(A) \x 'L_n(B) = 'L_n(G).
Proof.
move=> defG; have [[K H defA defB] _ _ tiAB] := dprodP defG.
rewrite !dprodEcp // in defG *; first exact: lcn_cprod.
by rewrite defA defB; apply/trivgP; rewrite -tiAB defA defB setISS ?lcn_sub.
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | lcn_dprod | |
der_cprodn A B G : A \* B = G -> A^`(n) \* B^`(n) = G^`(n).
Proof. by move=> defG; elim: n => {defG}// n; apply: (lcn_cprod 2). Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | der_cprod | |
der_dprodn A B G : A \x B = G -> A^`(n) \x B^`(n) = G^`(n).
Proof. by move=> defG; elim: n => {defG}// n; apply: (lcn_dprod 2). Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import fintype div bigop prime finset fingroup morphism",
"From mathcomp Require Import automorphism quotient commutator gproduct",
"From mathcomp Require Import perm gfunctor center gseries cyclic",
"From... | solvable/nilpotent.v | der_dprod |
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