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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 (subset_trans sHR) ?commSg.
pose m := #|G|.-1; exists m; first by rewrite mem_iota /= prednK.
set n := m; rewrite ['L__(G)]card_le1_trivg //= -(subnn m) -[m in _ - m]/n.
elim: n => [|n]; [by rewrite subn0 prednK | rewrite lcnSn subnS].
case: (eqsVneq 'L_n.+1(G) 1) => [-> | ntLn]; first by rewrite comm1G cards1.
case: (m - n) => [|m' /= IHn]; first by rewrite leqNgt cardG_gt1 ntLn.
rewrite -ltnS (leq_trans (proper_card _) IHn) //.
by rewrite (nil_comm_properl nilG) ?lcn_sub // subsetI subxx lcn_norm.
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 mathcomp Require finfun"
] |
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 nth_mkseq ?lcn_sub_leq.
right; apply/eqP/negPf; rewrite -(before_find 1 lt_n_c) nth_mkseq //.
exact: ltn_trans lt_n_c lt_c_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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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); rewrite ?abelian_nil //; 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 mathcomp Require finfun"
] |
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/trivgP.
case/cprodP: (lcn_cprod (m.+1 + n.+1) defGc) => _ <- _.
by rewrite mulG_subG /= -{1}LmH1 -LnK1 !lcn_sub_leq ?leq_addl ?leq_addr.
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 defG) nilA //= defB nilB.
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 / Z) * 'Z(K / Z) = 'Z(G / Z).
by rewrite -mulHK quotientMl // center_prod ?quotient_cents.
have ZquoZ (B A : {group gT}):
B \subset 'C(A) -> 'Z_n(A) * 'Z_n(B) = Z -> 'Z(A / Z) = 'Z_n.+1(A) / Z.
- move=> cAB {}defZ; have cAZnB: 'Z_n(B) \subset 'C(A) := gFsub_trans _ cAB.
have /second_isom[/=]: A \subset 'N(Z).
by rewrite -defZ normsM ?gFnorm ?cents_norm // centsC.
suffices ->: Z :&: A = 'Z_n(A).
by move=> f inj_f im_f; rewrite -!im_f ?gFsub // ucn_central injm_center.
rewrite -defZ -group_modl ?gFsub //; apply/mulGidPl.
have [-> | n_gt0] := posnP n; first exact: subsetIl.
by apply: subset_trans (ucn_sub_geq A n_gt0); rewrite /= setIC ucn1 setIS.
rewrite (ZquoZ H K) 1?centsC 1?(centC cZn) // {ZquoZ}(ZquoZ K H) //.
have cZn1: 'Z_n.+1(K) \subset 'C('Z_n.+1(H)) by apply: centSS cHK; apply: gFsub.
rewrite -quotientMl ?quotientK ?mul_subG ?gFsub_trans //=.
rewrite cprodE // -cent_joinEr ?mulSGid //= cent_joinEr //= -/Z.
by rewrite -defZ mulgSS ?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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 rewrite morphimS ?gFsub // quotient_cents2 ?gFsub_trans ?gFnorm_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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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_nil_classP _ _ (leqnn _)) //= injm_nil.
transitivity #|G|; apply/eqP; rewrite eqn_leq.
rewrite -(card_injm injf sGD) (leq_trans (index_size _ _)) ?size_mkseq //.
by rewrite leqNgt -nilpotent_class injm_nil ?nilG.
rewrite (leq_trans (index_size _ _)) ?size_mkseq // leqNgt -nilpotent_class.
by rewrite 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 mathcomp Require finfun"
] |
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 ?(ucn_sub_geq, leq_addl, comm_subG _ nZG, sub1set) //=.
by rewrite addSn /= ucnSnR inE.
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 ?ucn_sub_geq.
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 mathcomp Require finfun"
] |
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.
rewrite -subn1 leq_subLR addnC; apply/ucn_nil_classP => //=.
apply: (quotient_inj _ nsZ1G) => /=.
by apply: normalS (ucn_sub _ _) nsZ1G; rewrite /= addnS ucn_sub_geq.
by rewrite quotient_ucn_add; apply/ucn_nil_classP; rewrite //= quotient_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 mathcomp Require finfun"
] |
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 IHi] in nsGH *; first by rewrite sub1G in nsGH.
by case sZH: ('Z_i(G) \subset H); [exists i | apply: IHi; rewrite sZH].
apply: subset_trans sNH; rewrite subsetI ucn_sub -commg_subr.
by apply: subset_trans sZH; apply: subset_trans (ucn_comm i G); apply: 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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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_subnormal ?normalSG.
have sKG: K \subset G by rewrite subsetIl.
apply: subnormal_trans snHK (IHm _ (leq_trans _ leGHm) sKG).
by rewrite ltn_sub2l ?proper_card ?(proper_sub_trans prHK).
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 mathcomp Require finfun"
] |
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 -commg_subr commGC in nHG.
rewrite -IHn subsetI (subset_trans _ nHG) ?commSg ?subsetIl //=.
by rewrite (subset_trans _ (ucn_comm n G)) ?commSg ?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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 //.
apply: cyclic_factor_abelian (lcn_central 2 G) _.
by rewrite (isog_cyclic (third_isog _ _ _)) ?lcn_normal // 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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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; rewrite [G^`(_)]card_le1_trivg ?(leq_trans (IHn _)) ?subnn.
elim: n => [|n IHn]; first by rewrite subn0 prednK.
rewrite dergSn subnS -ltnS.
have [-> | ntGn] := eqVneq G^`(n) 1; first by rewrite commG1 cards1.
case: (_ - _) IHn => [|n']; first by rewrite leqNgt cardG_gt1 ntGn.
by apply: leq_trans (proper_card _); apply: sol_der1_proper (der_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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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_trans sKG nHG.
rewrite -quotient_sub1 // subG1 (forall_inP solGH) //.
by rewrite subsetI -morphimR ?morphimS.
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 mathcomp Require finfun"
] |
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 mathcomp Require finfun"
] |
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 prod_group_gT' := [the finGroupType of {dffun forall i, gT' i}].
pose f (x : prod_group_gT) : prod_group_gT' := [ffun i => x (lift ord0 i)].
have fm : morphic (setXn G) f.
apply/'forall_implyP => -[a b]; rewrite !inE/=.
by move=> /andP[/forallP aG /forallP bG]; apply/eqP/ffunP => i; rewrite !ffunE.
rewrite (@series_sol _ [group of setXn G] ('ker (morphm fm))) ?ker_normal//=.
rewrite (isog_sol (first_isog _))/=.
have -> : (morphm fm @* setXn G)%g = setXn (fun i => G (lift ord0 i)).
apply/setP => v; rewrite !inE morphimEdom; apply/idP/forallP => /=.
move=> /imsetP[/=x]; rewrite inE => /forallP/= xG ->.
by move=> i; rewrite morphmE ffunE xG.
move=> vG; apply/imsetP.
pose w := [ffun i : 'I_n.+1 =>
match unliftP ord0 i return (gT i) : Type with
| UnliftSome j i_eq => ecast i (gT i) (esym i_eq) (v j)
| UnliftNone i0 => 1%g
end].
have wl i : w (lift ord0 i) = v i.
rewrite ffunE; case: unliftP => //= j elij.
have eij : i = j by case: elij; apply/val_inj.
by rewrite [elij](eq_irrelevance _ (congr1 _ eij)); case: _ / eij.
have w0 : w ord0 = 1%g by rewrite ffunE; case: unliftP.
exists w; last by a
...
|
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 mathcomp Require finfun"
] |
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_mulr.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset prime fingroup morphism",
"From mathcomp Require Import gfunctor automorphism quotient action gproduct",
"From mathcomp Require Import cyclic"
] |
solvable/pgroup.v
|
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
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.