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 |
|---|---|---|---|---|---|---|
lcn_bigcprodn I r P (F : I -> {set gT}) G :
\big[cprod/1]_(i <- r | P i) F i = G ->
\big[cprod/1]_(i <- r | P i) 'L_n(F i) = 'L_n(G).
Proof.
elim/big_rec2: _ G => [_ <- | i A Z _ IH G dG]; first exact/esym/trivgP/lcn_sub.
by rewrite -(lcn_cprod n dG); have [[_ H _ dH]] := cprodP dG; rewrite dH (IH H).
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_bigcprod | |
lcn_bigdprodn I r P (F : I -> {set gT}) G :
\big[dprod/1]_(i <- r | P i) F i = G ->
\big[dprod/1]_(i <- r | P i) 'L_n(F i) = 'L_n(G).
Proof.
elim/big_rec2: _ G => [_ <- | i A Z _ IH G dG]; first exact/esym/trivgP/lcn_sub.
by rewrite -(lcn_dprod n dG); have [[_ H _ dH]] := dprodP dG; rewrite dH (IH H).
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_bigdprod | |
der_bigcprodn I r P (F : I -> {set gT}) G :
\big[cprod/1]_(i <- r | P i) F i = G ->
\big[cprod/1]_(i <- r | P i) (F i)^`(n) = G^`(n).
Proof.
elim/big_rec2: _ G => [_ <- | i A Z _ IH G dG]; first by rewrite gF1.
by rewrite -(der_cprod n dG); have [[_ H _ dH]] := cprodP dG; rewrite dH (IH H).
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_bigcprod | |
der_bigdprodn I r P (F : I -> {set gT}) G :
\big[dprod/1]_(i <- r | P i) F i = G ->
\big[dprod/1]_(i <- r | P i) (F i)^`(n) = G^`(n).
Proof.
elim/big_rec2: _ G => [_ <- | i A Z _ IH G dG]; first by rewrite gF1.
by rewrite -(der_dprod n dG); have [[_ H _ dH]] := dprodP dG; rewrite dH (IH H).
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_bigdprod | |
nilpotent_classG : nilpotent G = (nil_class G < #|G|).
Proof.
rewrite /nil_class; set s := mkseq _ _.
transitivity (1 \in s); last by rewrite -index_mem size_mkseq.
apply/idP/mapP=> {s}/= [nilG | [n _ Ln1]]; last first.
apply/forall_inP=> H /subsetIP[sHG sHR].
rewrite -subG1 {}Ln1; elim: n => // n IHn.
by 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 | nilpotent_class | |
lcn_nil_classPn G :
nilpotent G -> reflect ('L_n.+1(G) = 1) (nil_class G <= n).
Proof.
rewrite nilpotent_class /nil_class; set s := mkseq _ _.
set c := index 1 s => lt_c_G; case: leqP => [le_c_n | lt_n_c].
have Lc1: nth 1 s c = 1 by rewrite nth_index // -index_mem size_mkseq.
by left; apply/trivgP; rewrite -Lc1 n... | 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_nil_classP | |
lcnPG : reflect (exists n, 'L_n.+1(G) = 1) (nilpotent G).
Proof.
apply: (iffP idP) => [nilG | [n Ln1]].
by exists (nil_class G); apply/lcn_nil_classP.
apply/forall_inP=> H /subsetIP[sHG sHR]; rewrite -subG1 -{}Ln1.
by elim: n => // n IHn; rewrite (subset_trans sHR) ?commSg.
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 | lcnP | |
abelian_nilG : abelian G -> nilpotent G.
Proof. by move=> abG; apply/lcnP; exists 1%N; apply/commG1P. 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 | abelian_nil | |
nil_class0G : (nil_class G == 0) = (G :==: 1).
Proof.
apply/idP/eqP=> [nilG | ->].
by apply/(lcn_nil_classP 0); rewrite ?nilpotent_class (eqP nilG) ?cardG_gt0.
by rewrite -leqn0; apply/(lcn_nil_classP 0); rewrite ?nilpotent1.
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_class0 | |
nil_class1G : (nil_class G <= 1) = abelian G.
Proof.
have [-> | ntG] := eqsVneq G 1.
by rewrite abelian1 leq_eqVlt ltnS leqn0 nil_class0 eqxx orbT.
apply/idP/idP=> cGG.
apply/commG1P; apply/(lcn_nil_classP 1); rewrite // nilpotent_class.
by rewrite (leq_ltn_trans cGG) // cardG_gt1.
by apply/(lcn_nil_classP 1); re... | 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_class1 | |
cprod_nilA B G : A \* B = G -> nilpotent G = nilpotent A && nilpotent B.
Proof.
move=> defG; case/cprodP: defG (defG) => [[H K -> ->{A B}] defG _] defGc.
apply/idP/andP=> [nilG | [/lcnP[m LmH1] /lcnP[n LnK1]]].
by rewrite !(nilpotentS _ nilG) // -defG (mulG_subr, mulG_subl).
apply/lcnP; exists (m + n.+1); apply/trivg... | 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 | cprod_nil | |
mulg_nilG H :
H \subset 'C(G) -> nilpotent (G * H) = nilpotent G && nilpotent H.
Proof. by move=> cGH; rewrite -(cprod_nil (cprodEY cGH)) /= cent_joinEr. 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 | mulg_nil | |
dprod_nilA B G : A \x B = G -> nilpotent G = nilpotent A && nilpotent B.
Proof. by case/dprodP=> [[H K -> ->] <- cHK _]; rewrite mulg_nil.
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 | dprod_nil | |
bigdprod_nilI r (P : pred I) (A_ : I -> {set gT}) G :
\big[dprod/1]_(i <- r | P i) A_ i = G
-> (forall i, P i -> nilpotent (A_ i)) -> nilpotent G.
Proof.
move=> defG nilA; elim/big_rec: _ => [|i B Pi nilB] in G defG *.
by rewrite -defG nilpotent1.
have [[_ H _ defB] _ _ _] := dprodP defG.
by rewrite (dprod_nil de... | 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 | bigdprod_nil | |
lcn_contn : GFunctor.continuous (@lower_central_at n).
Proof.
case: n => //; elim=> // n IHn g0T h0T H phi.
by rewrite !lcnSn morphimR ?lcn_sub // commSg ?IHn.
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_cont | |
lcn_igFunn := [igFun by lcn_sub^~ n & lcn_cont n]. | 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 | lcn_igFun | |
lcn_gFunn := [gFun by lcn_cont n]. | 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 | lcn_gFun | |
lcn_mgFunn := [mgFun by fun _ G H => @lcnS _ n G H]. | 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 | lcn_mgFun | |
ucn_pmap: exists hZ : GFunctor.pmap, @upper_central_at n = hZ.
Proof.
elim: n => [|n' [hZ defZ]]; first by exists trivGfun_pgFun.
by exists [pgFun of @center %% hZ]; rewrite /= -defZ.
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 | ucn_pmap | |
ucn_group_setgT (G : {group gT}) : group_set 'Z_n(G).
Proof. by have [hZ ->] := ucn_pmap; 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 | ucn_group_set | |
upper_central_at_groupgT G := Group (@ucn_group_set gT 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 | upper_central_at_group | |
ucn_subgT (G : {group gT}) : 'Z_n(G) \subset G.
Proof. by have [hZ ->] := ucn_pmap; apply: gFsub. 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 | ucn_sub | |
morphim_ucn: GFunctor.pcontinuous (@upper_central_at n).
Proof. by have [hZ ->] := ucn_pmap; apply: pmorphimF. 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 | morphim_ucn | |
ucn_igFun:= [igFun by ucn_sub & morphim_ucn]. | 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 | ucn_igFun | |
ucn_gFun:= [gFun by morphim_ucn]. | 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 | ucn_gFun | |
ucn_pgFun:= [pgFun by morphim_ucn].
Variable (gT : finGroupType) (G : {group gT}). | 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 | ucn_pgFun | |
ucn_char: 'Z_n(G) \char G. Proof. exact: gFchar. 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 | ucn_char | |
ucn_norm: G \subset 'N('Z_n(G)). Proof. exact: gFnorm. 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 | ucn_norm | |
ucn_normal: 'Z_n(G) <| G. Proof. exact: gFnormal. 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 | ucn_normal | |
ucn0A : 'Z_0(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 | ucn0 | |
ucnSnn A : 'Z_n.+1(A) = coset 'Z_n(A) @*^-1 'Z(A / 'Z_n(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 | ucnSn | |
ucnEn A : 'Z_n(A) = iter n (fun B => coset B @*^-1 'Z(A / B)) 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 | ucnE | |
ucn_subSn G : 'Z_n(G) \subset 'Z_n.+1(G).
Proof. by rewrite -{1}['Z_n(G)]ker_coset morphpreS ?sub1G. 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 | ucn_subS | |
ucn_sub_geqm n G : n >= m -> 'Z_m(G) \subset 'Z_n(G).
Proof.
move/subnK <-; elim: {n}(n - m) => // n IHn.
exact: subset_trans (ucn_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 | ucn_sub_geq | |
ucn_centraln G : 'Z_n.+1(G) / 'Z_n(G) = 'Z(G / 'Z_n(G)).
Proof. by rewrite ucnSn cosetpreK. 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 | ucn_central | |
ucn_normalSn G : 'Z_n(G) <| 'Z_n.+1(G).
Proof. by rewrite (normalS _ _ (ucn_normal n G)) ?ucn_subS ?ucn_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 | ucn_normalS | |
ucn_commn G : [~: 'Z_n.+1(G), G] \subset 'Z_n(G).
Proof.
rewrite -quotient_cents2 ?normal_norm ?ucn_normal ?ucn_normalS //.
by rewrite ucn_central subsetIr.
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 | ucn_comm | |
ucn1G : 'Z_1(G) = 'Z(G).
Proof.
apply: (quotient_inj (normal1 _) (normal1 _)).
by rewrite /= (ucn_central 0) -injmF ?norms1 ?coset1_injm.
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 | ucn1 | |
ucnSnRn G : 'Z_n.+1(G) = [set x in G | [~: [set x], G] \subset 'Z_n(G)].
Proof. | 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 | ucnSnR | |
ucn_cprodn A B G : A \* B = G -> 'Z_n(A) \* 'Z_n(B) = 'Z_n(G).
Proof.
case/cprodP=> [[H K -> ->{A B}] mulHK cHK].
elim: n => [|n /cprodP[_ /= defZ cZn]]; first exact: cprod1g.
set Z := 'Z_n(G) in defZ cZn; rewrite (ucnSn n G) /= -/Z.
have /mulGsubP[nZH nZK]: H * K \subset 'N(Z) by rewrite mulHK gFnorm.
have <-: 'Z(H / ... | 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 | ucn_cprod | |
ucn_dprodn A B G : A \x B = G -> 'Z_n(A) \x 'Z_n(B) = 'Z_n(G).
Proof.
move=> defG; have [[K H defA defB] _ _ tiAB] := dprodP defG.
rewrite !dprodEcp // in defG *; first exact: ucn_cprod.
by rewrite defA defB; apply/trivgP; rewrite -tiAB defA defB setISS ?ucn_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 | ucn_dprod | |
ucn_bigcprodn I r P (F : I -> {set gT}) G :
\big[cprod/1]_(i <- r | P i) F i = G ->
\big[cprod/1]_(i <- r | P i) 'Z_n(F i) = 'Z_n(G).
Proof.
elim/big_rec2: _ G => [_ <- | i A Z _ IH G dG]; first by rewrite gF1.
by rewrite -(ucn_cprod n dG); have [[_ H _ dH]] := cprodP dG; rewrite dH (IH H).
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 | ucn_bigcprod | |
ucn_bigdprodn I r P (F : I -> {set gT}) G :
\big[dprod/1]_(i <- r | P i) F i = G ->
\big[dprod/1]_(i <- r | P i) 'Z_n(F i) = 'Z_n(G).
Proof.
elim/big_rec2: _ G => [_ <- | i A Z _ IH G dG]; first by rewrite gF1.
by rewrite -(ucn_dprod n dG); have [[_ H _ dH]] := dprodP dG; rewrite dH (IH H).
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 | ucn_bigdprod | |
ucn_lcnPn G : ('L_n.+1(G) == 1) = ('Z_n(G) == G).
Proof.
rewrite !eqEsubset sub1G ucn_sub /= andbT -(ucn0 G); set i := (n in LHS).
have: i + 0 = n by [rewrite addn0]; elim: i 0 => [j <- //|i IHi j].
rewrite addSnnS => /IHi <- {IHi}; rewrite ucnSn lcnSn.
rewrite -sub_morphim_pre ?gFsub_trans ?gFnorm_trans // subsetI.
by... | 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 | ucn_lcnP | |
ucnPG : reflect (exists n, 'Z_n(G) = G) (nilpotent G).
Proof.
apply: (iffP (lcnP G)) => -[n /eqP-clGn];
by exists n; apply/eqP; rewrite ucn_lcnP in clGn *.
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 | ucnP | |
ucn_nil_classPn G :
nilpotent G -> reflect ('Z_n(G) = G) (nil_class G <= n).
Proof.
move=> nilG; rewrite (sameP (lcn_nil_classP n nilG) eqP) ucn_lcnP; apply: eqP.
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 | ucn_nil_classP | |
ucn_idn G : 'Z_n('Z_n(G)) = 'Z_n(G).
Proof. exact: gFid. 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 | ucn_id | |
ucn_nilpotentn G : nilpotent 'Z_n(G).
Proof. by apply/ucnP; exists n; rewrite ucn_id. 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 | ucn_nilpotent | |
nil_class_ucnn G : nil_class 'Z_n(G) <= n.
Proof. by apply/ucn_nil_classP; rewrite ?ucn_nilpotent // ucn_id. 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_class_ucn | |
morphim_lcnn G : G \subset D -> f @* 'L_n(G) = 'L_n(f @* G).
Proof.
move=> sHG; case: n => //; elim=> // n IHn.
by rewrite !lcnSn -IHn morphimR // (subset_trans _ sHG) // 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 | morphim_lcn | |
injm_ucnn G : 'injm f -> G \subset D -> f @* 'Z_n(G) = 'Z_n(f @* G).
Proof. exact: injmF. 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 | injm_ucn | |
morphim_nilG : nilpotent G -> nilpotent (f @* G).
Proof.
case/ucnP=> n ZnG; apply/ucnP; exists n; apply/eqP.
by rewrite eqEsubset ucn_sub /= -{1}ZnG morphim_ucn.
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 | morphim_nil | |
injm_nilG : 'injm f -> G \subset D -> nilpotent (f @* G) = nilpotent G.
Proof.
move=> injf sGD; apply/idP/idP; last exact: morphim_nil.
case/ucnP=> n; rewrite -injm_ucn // => /injm_morphim_inj defZ.
by apply/ucnP; exists n; rewrite defZ ?gFsub_trans.
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 | injm_nil | |
nil_class_morphimG : nilpotent G -> nil_class (f @* G) <= nil_class G.
Proof.
move=> nilG; rewrite (sameP (ucn_nil_classP _ (morphim_nil nilG)) eqP) /=.
by rewrite eqEsubset ucn_sub -{1}(ucn_nil_classP _ nilG (leqnn _)) morphim_ucn.
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_class_morphim | |
nil_class_injmG :
'injm f -> G \subset D -> nil_class (f @* G) = nil_class G.
Proof.
move=> injf sGD; case nilG: (nilpotent G).
apply/eqP; rewrite eqn_leq nil_class_morphim //.
rewrite (sameP (lcn_nil_classP _ nilG) eqP) -subG1.
rewrite -(injmSK injf) ?gFsub_trans // morphim1.
by rewrite morphim_lcn // (lcn_n... | 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_class_injm | |
quotient_ucn_addm n G : 'Z_(m + n)(G) / 'Z_n(G) = 'Z_m(G / 'Z_n(G)).
Proof.
elim: m => [|m IHm]; first exact: trivg_quotient.
apply/setP=> Zx; have [x Nx ->{Zx}] := cosetP Zx.
have [sZG nZG] := andP (ucn_normal n G).
rewrite (ucnSnR m) inE -!sub1set -morphim_set1 //= -quotientR ?sub1set // -IHm.
rewrite !quotientSGK ?(... | 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 | quotient_ucn_add | |
isog_nilrT G (L : {group rT}) : G \isog L -> nilpotent G = nilpotent L.
Proof. by case/isogP=> f injf <-; rewrite injm_nil. 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 | isog_nil | |
isog_nil_classrT G (L : {group rT}) :
G \isog L -> nil_class G = nil_class L.
Proof. by case/isogP=> f injf <-; rewrite nil_class_injm. 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 | isog_nil_class | |
quotient_nilG H : nilpotent G -> nilpotent (G / H).
Proof. exact: morphim_nil. 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 | quotient_nil | |
quotient_center_nilG : nilpotent (G / 'Z(G)) = nilpotent G.
Proof.
rewrite -ucn1; apply/idP/idP; last exact: quotient_nil.
case/ucnP=> c nilGq; apply/ucnP; exists c.+1; have nsZ1G := ucn_normal 1 G.
apply: (quotient_inj _ nsZ1G); last by rewrite /= -(addn1 c) quotient_ucn_add.
by rewrite (normalS _ _ nsZ1G) ?ucn_sub ?u... | 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 | quotient_center_nil | |
nil_class_quotient_centerG :
nilpotent (G) -> nil_class (G / 'Z(G)) = (nil_class G).-1.
Proof.
move=> nilG; have nsZ1G := ucn_normal 1 G.
apply/eqP; rewrite -ucn1 eqn_leq; apply/andP; split.
apply/ucn_nil_classP; rewrite ?quotient_nil //= -quotient_ucn_add ucn1.
by rewrite (ucn_nil_classP _ _ _) ?addn1 ?leqSpred.... | 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_class_quotient_center | |
nilpotent_sub_normG H :
nilpotent G -> H \subset G -> 'N_G(H) \subset H -> G :=: H.
Proof.
move=> nilG sHG sNH; apply/eqP; rewrite eqEsubset sHG andbT; apply/negP=> nsGH.
have{nsGH} [i sZH []]: exists2 i, 'Z_i(G) \subset H & ~ 'Z_i.+1(G) \subset H.
case/ucnP: nilG => n ZnG; rewrite -{}ZnG in nsGH.
elim: n => [|i ... | 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 | nilpotent_sub_norm | |
nilpotent_proper_normG H :
nilpotent G -> H \proper G -> H \proper 'N_G(H).
Proof.
move=> nilG; rewrite properEneq properE subsetI normG => /andP[neHG sHG].
by rewrite sHG; apply: contra neHG => /(nilpotent_sub_norm nilG)->.
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 | nilpotent_proper_norm | |
nilpotent_subnormalG H : nilpotent G -> H \subset G -> H <|<| G.
Proof.
move=> nilG; have [m] := ubnP (#|G| - #|H|).
elim: m H => // m IHm H /ltnSE-leGHm sHG.
have [->|] := eqVproper sHG; first exact: subnormal_refl.
move/(nilpotent_proper_norm nilG); set K := 'N_G(H) => prHK.
have snHK: H <|<| K by rewrite normal_subn... | 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 | nilpotent_subnormal | |
TI_center_nilG H : nilpotent G -> H <| G -> H :&: 'Z(G) = 1 -> H :=: 1.
Proof.
move=> nilG /andP[sHG nHG] tiHZ.
rewrite -{1}(setIidPl sHG); have{nilG} /ucnP[n <-] := nilG.
elim: n => [|n IHn]; apply/trivgP; rewrite ?subsetIr // -tiHZ.
rewrite [H :&: 'Z(G)]setIA subsetI setIS ?ucn_sub //= (sameP commG1P trivgP).
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 | TI_center_nil | |
meet_center_nilG H :
nilpotent G -> H <| G -> H :!=: 1 -> H :&: 'Z(G) != 1.
Proof. by move=> nilG nsHG; apply: contraNneq => /TI_center_nil->. 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 | meet_center_nil | |
center_nil_eq1G : nilpotent G -> ('Z(G) == 1) = (G :==: 1).
Proof.
move=> nilG; apply/eqP/eqP=> [Z1 | ->]; last exact: center1.
by rewrite (TI_center_nil nilG) // (setIidPr (center_sub 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 | center_nil_eq1 | |
cyclic_nilpotent_quo_der1_cyclicG :
nilpotent G -> cyclic (G / G^`(1)) -> cyclic G.
Proof.
move=> nG; rewrite (isog_cyclic (quotient1_isog G)).
have [-> // | ntG' cGG'] := (eqVneq G^`(1) 1)%g.
suffices: 'L_2(G) \subset G :&: 'L_3(G) by move/(eqfun_inP nG)=> <-.
rewrite subsetI lcn_sub /= -quotient_cents2 ?lcn_norm //... | 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 | cyclic_nilpotent_quo_der1_cyclic | |
nilpotent_solG : nilpotent G -> solvable G.
Proof.
move=> nilG; apply/forall_inP=> H /subsetIP[sHG sHH'].
by rewrite (forall_inP nilG) // subsetI sHG (subset_trans sHH') ?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 | nilpotent_sol | |
abelian_solG : abelian G -> solvable G.
Proof. by move/abelian_nil/nilpotent_sol. 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 | abelian_sol | |
solvable1: solvable [1 gT]. Proof. exact: abelian_sol (abelian1 gT). 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 | solvable1 | |
solvableSG H : H \subset G -> solvable G -> solvable H.
Proof.
move=> sHG solG; apply/forall_inP=> K /subsetIP[sKH sKK'].
by rewrite (forall_inP solG) // subsetI (subset_trans sKH).
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 | solvableS | |
sol_der1_properG H :
solvable G -> H \subset G -> H :!=: 1 -> H^`(1) \proper H.
Proof.
move=> solG sHG ntH; rewrite properE comm_subG //; apply: implyP ntH.
by have:= forallP solG H; rewrite subsetI sHG implybNN.
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 | sol_der1_proper | |
derivedPG : reflect (exists n, G^`(n) = 1) (solvable G).
Proof.
apply: (iffP idP) => [solG | [n solGn]]; last first.
apply/forall_inP=> H /subsetIP[sHG sHH'].
rewrite -subG1 -{}solGn; elim: n => // n IHn.
exact: subset_trans sHH' (commgSS _ _).
suffices IHn n: #|G^`(n)| <= (#|G|.-1 - n).+1.
by exists #|G|.-1; r... | 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 | derivedP | |
morphim_sol: solvable G -> solvable (f @* G).
Proof.
move/(solvableS (subsetIr D G)); case/derivedP=> n Gn1; apply/derivedP.
by exists n; rewrite /= -morphimIdom -morphim_der ?subsetIl // Gn1 morphim1.
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 | morphim_sol | |
injm_sol: 'injm f -> G \subset D -> solvable (f @* G) = solvable G.
Proof.
move=> injf sGD; apply/idP/idP; last exact: morphim_sol.
case/derivedP=> n Gn1; apply/derivedP; exists n; apply/trivgP.
by rewrite -(injmSK injf) ?gFsub_trans ?morphim_der // Gn1 morphim1.
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 | injm_sol | |
isog_solG (L : {group rT}) : G \isog L -> solvable G = solvable L.
Proof. by case/isogP=> f injf <-; rewrite injm_sol. 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 | isog_sol | |
quotient_solG H : solvable G -> solvable (G / H).
Proof. exact: morphim_sol. 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 | quotient_sol | |
series_solG H : H <| G -> solvable G = solvable H && solvable (G / H).
Proof.
case/andP=> sHG nHG; apply/idP/andP=> [solG | [solH solGH]].
by rewrite quotient_sol // (solvableS sHG).
apply/forall_inP=> K /subsetIP[sKG sK'K].
suffices sKH: K \subset H by rewrite (forall_inP solH) // subsetI sKH.
have nHK := subset_tra... | 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 | series_sol | |
metacyclic_solG : metacyclic G -> solvable G.
Proof.
case/metacyclicP=> K [cycK nsKG cycGq].
by rewrite (series_sol nsKG) !abelian_sol ?cyclic_abelian.
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 | metacyclic_sol | |
setXn_soln (gT : 'I_n -> finGroupType) (G : forall i, {group gT i}) :
(forall i, solvable (G i)) -> solvable (setXn G).
Proof.
elim: n => [|n IHn] in gT G * => solG; first by rewrite groupX0 solvable1.
pose gT' (i : 'I_n) := gT (lift ord0 i).
pose prod_group_gT := [the finGroupType of {dffun forall i, gT i}].
pose pr... | 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 | setXn_sol | |
pgrouppi A := pi.-nat #|A|. | Definition | 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 | |
psubgrouppi A B := (B \subset A) && pgroup pi B. | Definition | 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 | psubgroup | |
p_groupA := pgroup (pdiv #|A|) A. | Definition | 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_group | |
p_eltpi x := pi.-nat #[x]. | Definition | 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 | |
consttx pi := x ^+ (chinese #[x]`_pi #[x]`_pi^' 1 0). | Definition | 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 | |
HallA B := (B \subset A) && coprime #|B| #|A : B|. | Definition | 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 | |
pHallpi A B := [&& B \subset A, pgroup pi B & pi^'.-nat #|A : B|]. | Definition | 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 | |
Sylp A := [set P : {group gT} | pHall p A P]. | Definition | 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 | Syl | |
SylowA B := p_group B && Hall A B. | Definition | 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 | Sylow | |
trivgVpdivG : G :=: 1 \/ (exists2 p, prime p & p %| #|G|).
Proof.
have [leG1|lt1G] := leqP #|G| 1; first by left; apply: card_le1_trivg.
by right; exists (pdiv #|G|); rewrite ?pdiv_dvd ?pdiv_prime.
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 | trivgVpdiv | |
prime_subgroupVtiG H : prime #|G| -> G \subset H \/ H :&: G = 1.
Proof.
move=> prG; have [|[p p_pr pG]] := trivgVpdiv (H :&: G); first by right.
left; rewrite (sameP setIidPr eqP) eqEcard subsetIr.
suffices <-: p = #|G| by rewrite dvdn_leq ?cardG_gt0.
by apply/eqP; rewrite -dvdn_prime2 // -(LagrangeI G H) setIC dvdn_mu... | 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 | prime_subgroupVti | |
pgroupEpi A : pi.-group A = pi.-nat #|A|. Proof. by []. 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 | pgroupE | |
sub_pgrouppi rho A : {subset pi <= rho} -> pi.-group A -> rho.-group A.
Proof. by move=> pi_sub_rho; apply: sub_in_pnat (in1W pi_sub_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 | sub_pgroup | |
eq_pgrouppi rho A : pi =i rho -> pi.-group A = rho.-group A.
Proof. exact: 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_pgroup | |
eq_p'grouppi rho A : pi =i rho -> pi^'.-group A = rho^'.-group A.
Proof. by move/eq_negn; 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'group | |
pgroupNKpi A : pi^'^'.-group A = pi.-group A.
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 | pgroupNK | |
pi_pgroupp pi A : p.-group A -> p \in pi -> pi.-group A.
Proof. exact: pi_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 | pi_pgroup | |
pi_p'groupp pi A : pi.-group A -> p \in pi^' -> p^'.-group A.
Proof. exact: pi_p'nat. 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 | pi_p'group | |
pi'_p'groupp pi A : pi^'.-group A -> p \in pi -> p^'.-group A.
Proof. exact: pi'_p'nat. 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 | pi'_p'group |
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