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
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|---|---|---|---|---|---|---|
Grp_modular_group:
'Mod_(p ^ n) \isog Grp (x : y : x ^+ q, y ^+ p, x ^ y = x ^+ r.+1).
Proof.
rewrite /modular_gtype def_p def_q def_r; apply: Extremal.Grp => //.
set B := <[_]>; have Bb: Zp1 \in B by apply: cycle_id.
have oB: #|B| = q by rewrite -orderE order_Zp1 Zp_cast.
have cycB: cyclic B by rewrite cycle_cyclic.
have pB: p.-group B by rewrite /pgroup oB pnatX ?pnat_id.
have ntB: B != 1 by rewrite -cardG_gt1 oB.
have [] := cyclic_pgroup_Aut_structure pB cycB ntB.
rewrite oB pfactorK //= -/B -(expg_znat r.+1 Bb) oB => mB [[def_mB _ _ _ _] _].
rewrite {1}def_n /= => [[t [At ot mBt]]].
have [p2 | ->] := even_prime p_pr; last first.
by case=> _ _ [s [As os mBs _]]; exists s; rewrite os -mBs def_mB.
rewrite {1}p2 /= -2!eqSS -addn2 -2!{1}subn1 -subnDA subnK 1?ltnW //.
case: eqP => [n3 _ | _ [_ [_ _ _ _ [s [As os mBs _ _]{t At ot mBt}]]]].
by exists t; rewrite At ot -def_mB // mBt /q /r p2 n3.
by exists s; rewrite As os -def_mB // mBs /r p2.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
Grp_modular_group
| |
modular_group_generatorsgT (xy : gT * gT) :=
let: (x, y) := xy in #[y] = p /\ x ^ y = x ^+ r.+1.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
modular_group_generators
| |
generators_modular_groupgT (G : {group gT}) :
G \isog 'Mod_m ->
exists2 xy, extremal_generators G p n xy & modular_group_generators xy.
Proof.
case/(isoGrpP _ Grp_modular_group); rewrite card_modular_group // -/m => oG.
case/existsP=> -[x y] /= /eqP[defG xq yp xy].
rewrite norm_joinEr ?norms_cycle ?xy ?mem_cycle // in defG.
have [Gx Gy]: x \in G /\ y \in G.
by apply/andP; rewrite -!cycle_subG -mulG_subG defG.
have notXy: y \notin <[x]>.
apply: contraL ltqm; rewrite -cycle_subG -oG -defG; move/mulGidPl->.
by rewrite -leqNgt dvdn_leq ?(ltnW q_gt1) // order_dvdn xq.
have oy: #[y] = p by apply: nt_prime_order (group1_contra notXy).
exists (x, y) => //=; split; rewrite ?inE ?notXy //.
apply/eqP; rewrite -(eqn_pmul2r p_gt0) -expnSr -{1}oy (ltn_predK n_gt2) -/m.
by rewrite -TI_cardMg ?defG ?oG // setIC prime_TIg ?cycle_subG // -orderE oy.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
generators_modular_group
| |
modular_group_structuregT (G : {group gT}) x y :
extremal_generators G p n (x, y) ->
G \isog 'Mod_m -> modular_group_generators (x, y) ->
let X := <[x]> in
[/\ [/\ X ><| <[y]> = G, ~~ abelian G
& {in X, forall z j, z ^ (y ^+ j) = z ^+ (j * r).+1}],
[/\ 'Z(G) = <[x ^+ p]>, 'Phi(G) = 'Z(G) & #|'Z(G)| = r],
[/\ G^`(1) = <[x ^+ r]>, #|G^`(1)| = p & nil_class G = 2],
forall k, k > 0 -> 'Mho^k(G) = <[x ^+ (p ^ k)]>
& if (p, n) == (2, 3) then 'Ohm_1(G) = G else
forall k, 0 < k < n.-1 ->
<[x ^+ (p ^ (n - k.+1))]> \x <[y]> = 'Ohm_k(G)
/\ #|'Ohm_k(G)| = (p ^ k.+1)%N].
Proof.
move=> genG isoG [oy xy] X.
have [oG Gx ox /setDP[Gy notXy]] := genG; rewrite -/m -/q in ox oG.
have [pG _ nsXG defXY nXY] := extremal_generators_facts p_pr genG.
have [sXG nXG] := andP nsXG; have sYG: <[y]> \subset G by rewrite cycle_subG.
have n1_gt1: n.-1 > 1 by [rewrite def_n]; have n1_gt0 := ltnW n1_gt1.
have def_n1 := prednK n1_gt0.
have def_m: (q * p)%N = m by rewrite -expnSr /m def_n.
have notcxy: y \notin 'C[x].
apply: contraL (introT eqP xy); move/cent1P=> cxy.
rewrite /conjg -cxy // eq_mulVg1 expgS !mulKg -order_dvdn ox.
by rewrite pfactor_dvdn ?expn_gt0 ?p_gt0 // pfactorK // -ltnNge prednK.
have tiXY: <[x]> :&: <[y]> = 1.
rewrite setIC prime_TIg -?orderE ?oy //; apply: contra notcxy.
by rewrite cycle_subG; apply: subsetP; rewrite cycle_subG cent1id.
have notcGG: ~~ abelian G.
by rewrite -defXY abelianM !cycle_abelian cent_cycle cycle_subG.
have cXpY: <
...
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
modular_group_structure
| |
card_ext_dihedral: #|ED| = (p./2 * m)%N.
Proof. by rewrite Extremal.card // /m -mul2n -divn2 mulnA divnK. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
card_ext_dihedral
| |
Grp_ext_dihedral: ED \isog Grp (x : y : x ^+ q, y ^+ p, x ^ y = x^-1).
Proof.
suffices isoED: ED \isog Grp (x : y : x ^+ q, y ^+ p, x ^ y = x ^+ q.-1).
move=> gT G; rewrite isoED.
apply: eq_existsb => [[x y]] /=; rewrite !xpair_eqE.
congr (_ && _); apply: andb_id2l; move/eqP=> xq1; congr (_ && (_ == _)).
by apply/eqP; rewrite eq_sym eq_invg_mul -expgS (ltn_predK q_gt1) xq1.
have unitrN1 : (- 1)%R \in GRing.unit by move=> R; rewrite unitrN unitr1.
pose uN1 := FinRing.unit ('Z_#[Zp1 : 'Z_q]) (unitrN1 _).
apply: Extremal.Grp => //; exists (Zp_unitm uN1).
rewrite Aut_aut order_injm ?injm_Zp_unitm ?in_setT //; split=> //.
by rewrite (dvdn_trans _ even_p) // order_dvdn -val_eqE /= mulrNN.
apply/eqP; rewrite autE ?cycle_id // eq_expg_mod_order /=.
by rewrite order_Zp1 !Zp_cast // !modn_mod (modn_small q_gt1) subn1.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
Grp_ext_dihedral
| |
card_dihedral: #|'D_m| = m.
Proof. by rewrite /('D_m)%type def_q card_ext_dihedral ?mul1n. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
card_dihedral
| |
Grp_dihedral: 'D_m \isog Grp (x : y : x ^+ q, y ^+ 2, x ^ y = x^-1).
Proof. by rewrite /('D_m)%type def_q; apply: Grp_ext_dihedral. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
Grp_dihedral
| |
Grp'_dihedral: 'D_m \isog Grp (x : y : x ^+ 2, y ^+ 2, (x * y) ^+ q).
Proof.
move=> gT G; rewrite Grp_dihedral; apply/existsP/existsP=> [] [[x y]] /=.
case/eqP=> <- xq1 y2 xy; exists (x * y, y); rewrite !xpair_eqE /= eqEsubset.
rewrite !join_subG !joing_subr !cycle_subG -{3}(mulgK y x) /=.
rewrite 2?groupM ?groupV ?mem_gen ?inE ?cycle_id ?orbT //= -mulgA expgS.
by rewrite {1}(conjgC x) xy -mulgA mulKg -(expgS y 1) y2 mulg1 xq1 !eqxx.
case/eqP=> <- x2 y2 xyq; exists (x * y, y); rewrite !xpair_eqE /= eqEsubset.
rewrite !join_subG !joing_subr !cycle_subG -{3}(mulgK y x) /=.
rewrite 2?groupM ?groupV ?mem_gen ?inE ?cycle_id ?orbT //= xyq y2 !eqxx /=.
by rewrite eq_sym eq_invg_mul !mulgA mulgK -mulgA -!(expgS _ 1) x2 y2 mulg1.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
Grp'_dihedral
| |
involutions_gen_dihedralgT (x y : gT) :
let G := <<[set x; y]>> in
#[x] = 2 -> #[y] = 2 -> x != y -> G \isog 'D_#|G|.
Proof.
move=> G ox oy ne_x_y; pose q := #[x * y].
have q_gt1: q > 1 by rewrite order_gt1 -eq_invg_mul invg_expg ox.
have homG: G \homg 'D_q.*2.
rewrite Grp'_dihedral //; apply/existsP; exists (x, y); rewrite /= !xpair_eqE.
by rewrite joing_idl joing_idr -{1}ox -oy !expg_order !eqxx.
suff oG: #|G| = q.*2 by rewrite oG isogEcard oG card_dihedral ?leqnn ?andbT.
have: #|G| %| q.*2 by rewrite -card_dihedral ?card_homg.
have Gxy: <[x * y]> \subset G.
by rewrite cycle_subG groupM ?mem_gen ?set21 ?set22.
have[k oG]: exists k, #|G| = (k * q)%N by apply/dvdnP; rewrite cardSg.
rewrite oG -mul2n dvdn_pmul2r ?order_gt0 ?dvdn_divisors // !inE /=.
case/pred2P=> [k1 | -> //]; case/negP: ne_x_y.
have cycG: cyclic G.
apply/cyclicP; exists (x * y); apply/eqP.
by rewrite eq_sym eqEcard Gxy oG k1 mul1n leqnn.
have: <[x]> == <[y]>.
by rewrite (eq_subG_cyclic cycG) ?genS ?subsetUl ?subsetUr -?orderE ?ox ?oy.
by rewrite eqEcard cycle_subG /= cycle2g // !inE -order_eq1 ox; case/andP.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
involutions_gen_dihedral
| |
Grp_2dihedraln : n > 1 ->
'D_(2 ^ n) \isog Grp (x : y : x ^+ (2 ^ n.-1), y ^+ 2, x ^ y = x^-1).
Proof.
move=> n_gt1; rewrite -(ltn_predK n_gt1) expnS mul2n /=.
by apply: Grp_dihedral; rewrite (ltn_exp2l 0) // -(subnKC n_gt1).
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
Grp_2dihedral
| |
card_2dihedraln : n > 1 -> #|'D_(2 ^ n)| = (2 ^ n)%N.
Proof.
move=> n_gt1; rewrite -(ltn_predK n_gt1) expnS mul2n /= card_dihedral //.
by rewrite (ltn_exp2l 0) // -(subnKC n_gt1).
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
card_2dihedral
| |
card_semidihedraln : n > 3 -> #|'SD_(2 ^ n)| = (2 ^ n)%N.
Proof.
move=> n_gt3.
rewrite /('SD__)%type -(subnKC (ltnW (ltnW n_gt3))) pdiv_pfactor //.
by rewrite // !expnS !mulKn -?expnS ?Extremal.card //= (ltn_exp2l 0).
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
card_semidihedral
| |
Grp_semidihedraln : n > 3 ->
'SD_(2 ^ n) \isog
Grp (x : y : x ^+ (2 ^ n.-1), y ^+ 2, x ^ y = x ^+ (2 ^ n.-2).-1).
Proof.
move=> n_gt3.
rewrite /('SD__)%type -(subnKC (ltnW (ltnW n_gt3))) pdiv_pfactor //.
rewrite !expnS !mulKn // -!expnS /=; set q := (2 ^ _)%N.
have q_gt1: q > 1 by rewrite (ltn_exp2l 0).
apply: Extremal.Grp => //; set B := <[_]>.
have oB: #|B| = q by rewrite -orderE order_Zp1 Zp_cast.
have pB: 2.-group B by rewrite /pgroup oB pnatX.
have ntB: B != 1 by rewrite -cardG_gt1 oB.
have [] := cyclic_pgroup_Aut_structure pB (cycle_cyclic _) ntB.
rewrite oB /= pfactorK //= -/B => m [[def_m _ _ _ _] _].
rewrite -{1 2}(subnKC n_gt3) => [[t [At ot _ [s [_ _ _ defA]]]]].
case/dprodP: defA => _ defA cst _.
have{cst defA} cAt: t \in 'C(Aut B).
rewrite -defA centM inE -sub_cent1 -cent_cycle centsC cst /=.
by rewrite cent_cycle cent1id.
case=> s0 [As0 os0 _ def_s0t _]; exists (s0 * t).
rewrite -def_m ?groupM ?cycle_id // def_s0t !Zp_expg !mul1n valZpK Zp_nat.
rewrite order_dvdn expgMn /commute 1?(centP cAt) // -{1}os0 -{1}ot.
by rewrite !expg_order mul1g.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
Grp_semidihedral
| |
card_quaternion: #|'Q_m| = m. Proof. by case defQ. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
card_quaternion
| |
Grp_quaternion: GrpQ. Proof. by case defQ. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
Grp_quaternion
| |
eq_Mod8_D8: 'Mod_8 = 'D_8. Proof. by []. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
eq_Mod8_D8
| |
generators_2dihedral:
n > 1 -> G \isog 'D_m ->
exists2 xy, extremal_generators G 2 n xy
& let: (x, y) := xy in #[y] = 2 /\ x ^ y = x^-1.
Proof.
move=> n_gt1; have [def2q _ ltqm _] := def2qr n_gt1.
case/(isoGrpP _ (Grp_2dihedral n_gt1)); rewrite card_2dihedral // -/ m => oG.
case/existsP=> -[x y] /=; rewrite -/q => /eqP[defG xq y2 xy].
have{} defG: <[x]> * <[y]> = G.
by rewrite -norm_joinEr // norms_cycle xy groupV cycle_id.
have notXy: y \notin <[x]>.
apply: contraL ltqm => Xy; rewrite -leqNgt -oG -defG mulGSid ?cycle_subG //.
by rewrite dvdn_leq // order_dvdn xq.
have oy: #[y] = 2 by apply: nt_prime_order (group1_contra notXy).
have ox: #[x] = q.
apply: double_inj; rewrite -muln2 -oy -mul2n def2q -oG -defG TI_cardMg //.
by rewrite setIC prime_TIg ?cycle_subG // -orderE oy.
exists (x, y) => //=.
by rewrite oG ox !inE notXy -!cycle_subG /= -defG mulG_subl mulG_subr.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
generators_2dihedral
| |
generators_semidihedral:
n > 3 -> G \isog 'SD_m ->
exists2 xy, extremal_generators G 2 n xy
& let: (x, y) := xy in #[y] = 2 /\ x ^ y = x ^+ r.-1.
Proof.
move=> n_gt3; have [def2q _ ltqm _] := def2qr (ltnW (ltnW n_gt3)).
case/(isoGrpP _ (Grp_semidihedral n_gt3)).
rewrite card_semidihedral // -/m => oG.
case/existsP=> -[x y] /=; rewrite -/q -/r => /eqP[defG xq y2 xy].
have{} defG: <[x]> * <[y]> = G.
by rewrite -norm_joinEr // norms_cycle xy mem_cycle.
have notXy: y \notin <[x]>.
apply: contraL ltqm => Xy; rewrite -leqNgt -oG -defG mulGSid ?cycle_subG //.
by rewrite dvdn_leq // order_dvdn xq.
have oy: #[y] = 2 by apply: nt_prime_order (group1_contra notXy).
have ox: #[x] = q.
apply: double_inj; rewrite -muln2 -oy -mul2n def2q -oG -defG TI_cardMg //.
by rewrite setIC prime_TIg ?cycle_subG // -orderE oy.
exists (x, y) => //=.
by rewrite oG ox !inE notXy -!cycle_subG /= -defG mulG_subl mulG_subr.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
generators_semidihedral
| |
generators_quaternion:
n > 2 -> G \isog 'Q_m ->
exists2 xy, extremal_generators G 2 n xy
& let: (x, y) := xy in [/\ #[y] = 4, y ^+ 2 = x ^+ r & x ^ y = x^-1].
Proof.
move=> n_gt2; have [def2q def2r ltqm _] := def2qr (ltnW n_gt2).
case/(isoGrpP _ (Grp_quaternion n_gt2)); rewrite card_quaternion // -/m => oG.
case/existsP=> -[x y] /=; rewrite -/q -/r => /eqP[defG xq y2 xy].
have{} defG: <[x]> * <[y]> = G.
by rewrite -norm_joinEr // norms_cycle xy groupV cycle_id.
have notXy: y \notin <[x]>.
apply: contraL ltqm => Xy; rewrite -leqNgt -oG -defG mulGSid ?cycle_subG //.
by rewrite dvdn_leq // order_dvdn xq.
have ox: #[x] = q.
apply/eqP; rewrite eqn_leq dvdn_leq ?order_dvdn ?xq //=.
rewrite -(leq_pmul2r (order_gt0 y)) mul_cardG defG oG -def2q mulnAC mulnC.
rewrite leq_pmul2r // dvdn_leq ?muln_gt0 ?cardG_gt0 // order_dvdn expgM.
by rewrite -order_dvdn order_dvdG //= inE {1}y2 !mem_cycle.
have oy2: #[y ^+ 2] = 2 by rewrite y2 orderXdiv ox -def2r ?dvdn_mull ?mulnK.
exists (x, y) => /=; last by rewrite (orderXprime oy2).
by rewrite oG !inE notXy -!cycle_subG /= -defG mulG_subl mulG_subr.
Qed.
Variables x y : gT.
Implicit Type M : {group gT}.
Let X := <[x]>.
Let Y := <[y]>.
Let yG := y ^: G.
Let xyG := (x * y) ^: G.
Let My := <<yG>>.
Let Mxy := <<xyG>>.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
generators_quaternion
| |
dihedral2_structure:
n > 1 -> extremal_generators G 2 n (x, y) -> G \isog 'D_m ->
[/\ [/\ X ><| Y = G, {in G :\: X, forall t, #[t] = 2}
& {in X & G :\: X, forall z t, z ^ t = z^-1}],
[/\ G ^`(1) = <[x ^+ 2]>, 'Phi(G) = G ^`(1), #|G^`(1)| = r
& nil_class G = n.-1],
'Ohm_1(G) = G /\ (forall k, k > 0 -> 'Mho^k(G) = <[x ^+ (2 ^ k)]>),
[/\ yG :|: xyG = G :\: X, [disjoint yG & xyG]
& forall M, maximal M G = pred3 X My Mxy M]
& if n == 2 then (2.-abelem G : Prop) else
[/\ 'Z(G) = <[x ^+ r]>, #|'Z(G)| = 2,
My \isog 'D_q, Mxy \isog 'D_q
& forall U, cyclic U -> U \subset G -> #|G : U| = 2 -> U = X]].
Proof.
move=> n_gt1 genG isoG; have [def2q def2r ltqm ltrq] := def2qr n_gt1.
have [oG Gx ox X'y] := genG; rewrite -/m -/q -/X in oG ox X'y.
case/extremal_generators_facts: genG; rewrite -/X // => pG maxX nsXG defXY nXY.
have [sXG nXG]:= andP nsXG; have [Gy notXy]:= setDP X'y.
have ox2: #[x ^+ 2] = r by rewrite orderXdiv ox -def2r ?dvdn_mulr ?mulKn.
have oxr: #[x ^+ r] = 2 by rewrite orderXdiv ox -def2r ?dvdn_mull ?mulnK.
have [[u v] [_ Gu ou U'v] [ov uv]] := generators_2dihedral n_gt1 isoG.
have defUv: <[u]> :* v = G :\: <[u]>.
apply: rcoset_index2; rewrite -?divgS ?cycle_subG //.
by rewrite oG -orderE ou -def2q mulnK.
have invUV: {in <[u]> & <[u]> :* v, forall z t, z ^ t = z^-1}.
move=> z t; case/cycleP=> i ->; case/rcosetP=> z'; case/cycleP=> j -> ->{z t}.
by rewrite conjgM {2}/conjg commuteX2 // mulKg conjXg uv expgVn.
have oU': {in
...
|
Theorem
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
dihedral2_structure
| |
quaternion_structure:
n > 2 -> extremal_generators G 2 n (x, y) -> G \isog 'Q_m ->
[/\ [/\ pprod X Y = G, {in G :\: X, forall t, #[t] = 4}
& {in X & G :\: X, forall z t, z ^ t = z^-1}],
[/\ G ^`(1) = <[x ^+ 2]>, 'Phi(G) = G ^`(1), #|G^`(1)| = r
& nil_class G = n.-1],
[/\ 'Z(G) = <[x ^+ r]>, #|'Z(G)| = 2,
forall u, u \in G -> #[u] = 2 -> u = x ^+ r,
'Ohm_1(G) = <[x ^+ r]> /\ 'Ohm_2(G) = G
& forall k, k > 0 -> 'Mho^k(G) = <[x ^+ (2 ^ k)]>],
[/\ yG :|: xyG = G :\: X /\ [disjoint yG & xyG]
& forall M, maximal M G = pred3 X My Mxy M]
& n > 3 ->
[/\ My \isog 'Q_q, Mxy \isog 'Q_q
& forall U, cyclic U -> U \subset G -> #|G : U| = 2 -> U = X]].
Proof.
move=> n_gt2 genG isoG; have [def2q def2r ltqm ltrq] := def2qr (ltnW n_gt2).
have [oG Gx ox X'y] := genG; rewrite -/m -/q -/X in oG ox X'y.
case/extremal_generators_facts: genG; rewrite -/X // => pG maxX nsXG defXY nXY.
have [sXG nXG]:= andP nsXG; have [Gy notXy]:= setDP X'y.
have oxr: #[x ^+ r] = 2 by rewrite orderXdiv ox -def2r ?dvdn_mull ?mulnK.
have ox2: #[x ^+ 2] = r by rewrite orderXdiv ox -def2r ?dvdn_mulr ?mulKn.
have [[u v] [_ Gu ou U'v] [ov v2 uv]] := generators_quaternion n_gt2 isoG.
have defUv: <[u]> :* v = G :\: <[u]>.
apply: rcoset_index2; rewrite -?divgS ?cycle_subG //.
by rewrite oG -orderE ou -def2q mulnK.
have invUV: {in <[u]> & <[u]> :* v, forall z t, z ^ t = z^-1}.
move=> z t; case/cycleP=> i ->; case/rcosetP=> ?; case/cycleP=> j -> ->{z t}
...
|
Theorem
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
quaternion_structure
| |
semidihedral_structure:
n > 3 -> extremal_generators G 2 n (x, y) -> G \isog 'SD_m -> #[y] = 2 ->
[/\ [/\ X ><| Y = G, #[x * y] = 4
& {in X & G :\: X, forall z t, z ^ t = z ^+ r.-1}],
[/\ G ^`(1) = <[x ^+ 2]>, 'Phi(G) = G ^`(1), #|G^`(1)| = r
& nil_class G = n.-1],
[/\ 'Z(G) = <[x ^+ r]>, #|'Z(G)| = 2,
'Ohm_1(G) = My /\ 'Ohm_2(G) = G
& forall k, k > 0 -> 'Mho^k(G) = <[x ^+ (2 ^ k)]>],
[/\ yG :|: xyG = G :\: X /\ [disjoint yG & xyG]
& forall H, maximal H G = pred3 X My Mxy H]
& [/\ My \isog 'D_q, Mxy \isog 'Q_q
& forall U, cyclic U -> U \subset G -> #|G : U| = 2 -> U = X]].
Proof.
move=> n_gt3 genG isoG oy.
have [def2q def2r ltqm ltrq] := def2qr (ltnW (ltnW n_gt3)).
have [oG Gx ox X'y] := genG; rewrite -/m -/q -/X in oG ox X'y.
case/extremal_generators_facts: genG; rewrite -/X // => pG maxX nsXG defXY nXY.
have [sXG nXG]:= andP nsXG; have [Gy notXy]:= setDP X'y.
have ox2: #[x ^+ 2] = r by rewrite orderXdiv ox -def2r ?dvdn_mulr ?mulKn.
have oxr: #[x ^+ r] = 2 by rewrite orderXdiv ox -def2r ?dvdn_mull ?mulnK.
have [[u v] [_ Gu ou U'v] [ov uv]] := generators_semidihedral n_gt3 isoG.
have defUv: <[u]> :* v = G :\: <[u]>.
apply: rcoset_index2; rewrite -?divgS ?cycle_subG //.
by rewrite oG -orderE ou -def2q mulnK.
have invUV: {in <[u]> & <[u]> :* v, forall z t, z ^ t = z ^+ r.-1}.
move=> z t; case/cycleP=> i ->; case/rcosetP=> ?; case/cycleP=> j -> ->{z t}.
by rewrite conjgM {2}/conjg commuteX2 // mulKg conjXg uv -!expg
...
|
Theorem
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
semidihedral_structure
| |
extremal_group_type:=
ModularGroup | Dihedral | SemiDihedral | Quaternion | NotExtremal.
|
Inductive
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
extremal_group_type
| |
index_extremal_group_typec : nat :=
match c with
| ModularGroup => 0
| Dihedral => 1
| SemiDihedral => 2
| Quaternion => 3
| NotExtremal => 4
end.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
index_extremal_group_type
| |
enum_extremal_groups:=
[:: ModularGroup; Dihedral; SemiDihedral; Quaternion].
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
enum_extremal_groups
| |
cancel_index_extremal_groups:
cancel index_extremal_group_type (nth NotExtremal enum_extremal_groups).
Proof. by case. Qed.
Local Notation extgK := cancel_index_extremal_groups.
#[export]
HB.instance Definition _ := Countable.copy extremal_group_type (can_type extgK).
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
cancel_index_extremal_groups
| |
bound_extremal_groups(c : extremal_group_type) : pickle c < 6.
Proof. by case: c. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
bound_extremal_groups
| |
extremal_class(A : {set gT}) :=
let m := #|A| in let p := pdiv m in let n := logn p m in
if (n > 1) && (A \isog 'D_(2 ^ n)) then Dihedral else
if (n > 2) && (A \isog 'Q_(2 ^ n)) then Quaternion else
if (n > 3) && (A \isog 'SD_(2 ^ n)) then SemiDihedral else
if (n > 2) && (A \isog 'Mod_(p ^ n)) then ModularGroup else
NotExtremal.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
extremal_class
| |
extremal2A := extremal_class A \in behead enum_extremal_groups.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
extremal2
| |
dihedral_classP:
extremal_class G = Dihedral <-> (exists2 n, n > 1 & G \isog 'D_(2 ^ n)).
Proof.
rewrite /extremal_class; split=> [ | [n n_gt1 isoG]].
by move: (logn _ _) => n; do 4?case: ifP => //; case/andP; exists n.
rewrite (card_isog isoG) card_2dihedral // -(ltn_predK n_gt1) pdiv_pfactor //.
by rewrite pfactorK // (ltn_predK n_gt1) n_gt1 isoG.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
dihedral_classP
| |
quaternion_classP:
extremal_class G = Quaternion <-> (exists2 n, n > 2 & G \isog 'Q_(2 ^ n)).
Proof.
rewrite /extremal_class; split=> [ | [n n_gt2 isoG]].
by move: (logn _ _) => n; do 4?case: ifP => //; case/andP; exists n.
rewrite (card_isog isoG) card_quaternion // -(ltn_predK n_gt2) pdiv_pfactor //.
rewrite pfactorK // (ltn_predK n_gt2) n_gt2 isoG.
case: andP => // [[n_gt1 isoGD]].
have [[x y] genG [oy _ _]]:= generators_quaternion n_gt2 isoG.
have [_ _ _ X'y] := genG.
by case/dihedral2_structure: genG oy => // [[_ ->]].
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
quaternion_classP
| |
semidihedral_classP:
extremal_class G = SemiDihedral <-> (exists2 n, n > 3 & G \isog 'SD_(2 ^ n)).
Proof.
rewrite /extremal_class; split=> [ | [n n_gt3 isoG]].
by move: (logn _ _) => n; do 4?case: ifP => //; case/andP; exists n.
rewrite (card_isog isoG) card_semidihedral //.
rewrite -(ltn_predK n_gt3) pdiv_pfactor // pfactorK // (ltn_predK n_gt3) n_gt3.
have [[x y] genG [oy _]]:= generators_semidihedral n_gt3 isoG.
have [_ Gx _ X'y]:= genG.
case: andP => [[n_gt1 isoGD]|_].
have [[_ oxy _ _] _ _ _]:= semidihedral_structure n_gt3 genG isoG oy.
case: (dihedral2_structure n_gt1 genG isoGD) oxy => [[_ ->]] //.
by rewrite !inE !groupMl ?cycle_id in X'y *.
case: andP => // [[n_gt2 isoGQ]|]; last by rewrite isoG.
by case: (quaternion_structure n_gt2 genG isoGQ) oy => [[_ ->]].
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
semidihedral_classP
| |
odd_not_extremal2: odd #|G| -> ~~ extremal2 G.
Proof.
rewrite /extremal2 /extremal_class; case: logn => // n'.
case: andP => [[n_gt1 isoG] | _].
by rewrite (card_isog isoG) card_2dihedral ?oddX.
case: andP => [[n_gt2 isoG] | _].
by rewrite (card_isog isoG) card_quaternion ?oddX.
case: andP => [[n_gt3 isoG] | _].
by rewrite (card_isog isoG) card_semidihedral ?oddX.
by case: ifP.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
odd_not_extremal2
| |
modular_group_classP:
extremal_class G = ModularGroup
<-> (exists2 p, prime p &
exists2 n, n >= (p == 2) + 3 & G \isog 'Mod_(p ^ n)).
Proof.
rewrite /extremal_class; split=> [ | [p p_pr [n n_gt23 isoG]]].
move: (pdiv _) => p; set n := logn p _; do 4?case: ifP => //.
case/andP=> n_gt2 isoG _ _; rewrite ltnW //= => not_isoG _.
exists p; first by move: n_gt2; rewrite /n lognE; case (prime p).
exists n => //; case: eqP => // p2; rewrite ltn_neqAle; case: eqP => // n3.
by case/idP: not_isoG; rewrite p2 -n3 in isoG *.
have n_gt2 := leq_trans (leq_addl _ _) n_gt23; have n_gt1 := ltnW n_gt2.
have n_gt0 := ltnW n_gt1; have def_n := prednK n_gt0.
have [[x y] genG mod_xy] := generators_modular_group p_pr n_gt2 isoG.
case/modular_group_structure: (genG) => // _ _ [_ _ nil2G] _ _.
have [oG _ _ _] := genG; have [oy _] := mod_xy.
rewrite oG -def_n pdiv_pfactor // def_n pfactorK // n_gt1 n_gt2 {}isoG /=.
case: (ltngtP p 2) => [|p_gt2|p2]; first by rewrite ltnNge prime_gt1.
rewrite !(isog_sym G) !isogEcard card_2dihedral ?card_quaternion //= oG.
rewrite leq_exp2r // leqNgt p_gt2 !andbF; case: and3P=> // [[n_gt3 _]].
by rewrite card_semidihedral // leq_exp2r // leqNgt p_gt2.
rewrite p2 in genG oy n_gt23; rewrite n_gt23.
have: nil_class G <> n.-1.
by apply/eqP; rewrite neq_ltn -ltnS nil2G def_n n_gt23.
case: ifP => [isoG | _]; first by case/dihedral2_structure: genG => // _ [].
case: ifP => [isoG | _]; first by case/quaternion_structure: genG => // _ [].
by case: ifP => // isoG;
...
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
modular_group_classP
| |
extremal2_structure(gT : finGroupType) (G : {group gT}) n x y :
let cG := extremal_class G in
let m := (2 ^ n)%N in let q := (2 ^ n.-1)%N in let r := (2 ^ n.-2)%N in
let X := <[x]> in let yG := y ^: G in let xyG := (x * y) ^: G in
let My := <<yG>> in let Mxy := <<xyG>> in
extremal_generators G 2 n (x, y) ->
extremal2 G -> (cG == SemiDihedral) ==> (#[y] == 2) ->
[/\ [/\ (if cG == Quaternion then pprod X <[y]> else X ><| <[y]>) = G,
if cG == SemiDihedral then #[x * y] = 4 else
{in G :\: X, forall z, #[z] = (if cG == Dihedral then 2 else 4)},
if cG != Quaternion then True else
{in G, forall z, #[z] = 2 -> z = x ^+ r}
& {in X & G :\: X, forall t z,
t ^ z = (if cG == SemiDihedral then t ^+ r.-1 else t^-1)}],
[/\ G ^`(1) = <[x ^+ 2]>, 'Phi(G) = G ^`(1), #|G^`(1)| = r
& nil_class G = n.-1],
[/\ if n > 2 then 'Z(G) = <[x ^+ r]> /\ #|'Z(G)| = 2 else 2.-abelem G,
'Ohm_1(G) = (if cG == Quaternion then <[x ^+ r]> else
if cG == SemiDihedral then My else G),
'Ohm_2(G) = G
& forall k, k > 0 -> 'Mho^k(G) = <[x ^+ (2 ^ k)]>],
[/\ yG :|: xyG = G :\: X, [disjoint yG & xyG]
& forall H : {group gT}, maximal H G = (gval H \in pred3 X My Mxy)]
& if n <= (cG == Quaternion) + 2 then True else
[/\ forall U, cyclic U -> U \subset G -> #|G : U| = 2 -> U = X,
if cG == Quaternion then My \isog 'Q_q else My \isog 'D_q,
extremal_class My = (if
...
|
Theorem
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
extremal2_structure
| |
maximal_cycle_extremalgT p (G X : {group gT}) :
p.-group G -> ~~ abelian G -> cyclic X -> X \subset G -> #|G : X| = p ->
(extremal_class G == ModularGroup) || (p == 2) && extremal2 G.
Proof.
move=> pG not_cGG cycX sXG iXG; rewrite /extremal2; set cG := extremal_class G.
have [|p_pr _ _] := pgroup_pdiv pG.
by case: eqP not_cGG => // ->; rewrite abelian1.
have p_gt1 := prime_gt1 p_pr; have p_gt0 := ltnW p_gt1.
have [n oG] := p_natP pG; have n_gt2: n > 2.
apply: contraR not_cGG; rewrite -leqNgt => n_le2.
by rewrite (p2group_abelian pG) // oG pfactorK.
have def_n := subnKC n_gt2; have n_gt1 := ltnW n_gt2; have n_gt0 := ltnW n_gt1.
pose q := (p ^ n.-1)%N; pose r := (p ^ n.-2)%N.
have q_gt1: q > 1 by rewrite (ltn_exp2l 0) // -(subnKC n_gt2).
have r_gt0: r > 0 by rewrite expn_gt0 p_gt0.
have def_pr: (p * r)%N = q by rewrite /q /r -def_n.
have oX: #|X| = q by rewrite -(divg_indexS sXG) oG iXG /q -def_n mulKn.
have ntX: X :!=: 1 by rewrite -cardG_gt1 oX.
have maxX: maximal X G by rewrite p_index_maximal ?iXG.
have nsXG: X <| G := p_maximal_normal pG maxX; have [_ nXG] := andP nsXG.
have cXX: abelian X := cyclic_abelian cycX.
have scXG: 'C_G(X) = X.
apply/eqP; rewrite eqEsubset subsetI sXG -abelianE cXX !andbT.
apply: contraR not_cGG; case/subsetPn=> y; case/setIP=> Gy cXy notXy.
rewrite -!cycle_subG in Gy notXy; rewrite -(mulg_normal_maximal nsXG _ Gy) //.
by rewrite abelianM cycle_abelian cyclic_abelian ?cycle_subG.
have [x defX] := cyclicP cycX; have pX := pgroupS sXG pG.
have Xx: x \
...
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
maximal_cycle_extremal
| |
cyclic_SCNgT p (G U : {group gT}) :
p.-group G -> U \in 'SCN(G) -> ~~ abelian G -> cyclic U ->
[/\ p = 2, #|G : U| = 2 & extremal2 G]
\/ exists M : {group gT},
[/\ M :=: 'C_G('Mho^1(U)), #|M : U| = p, extremal_class M = ModularGroup,
'Ohm_1(M)%G \in 'E_p^2(G) & 'Ohm_1(M) \char G].
Proof.
move=> pG /SCN_P[nsUG scUG] not_cGG cycU; have [sUG nUG] := andP nsUG.
have [cUU pU] := (cyclic_abelian cycU, pgroupS sUG pG).
have ltUG: ~~ (G \subset U).
by apply: contra not_cGG => sGU; apply: abelianS cUU.
have ntU: U :!=: 1.
by apply: contraNneq ltUG => U1; rewrite -scUG subsetIidl U1 cents1.
have [p_pr _ [n oU]] := pgroup_pdiv pU ntU.
have p_gt1 := prime_gt1 p_pr; have p_gt0 := ltnW p_gt1.
have [u defU] := cyclicP cycU; have Uu: u \in U by rewrite defU cycle_id.
have Gu := subsetP sUG u Uu; have p_u := mem_p_elt pG Gu.
have defU1: 'Mho^1(U) = <[u ^+ p]> by rewrite defU (Mho_p_cycle _ p_u).
have modM1 (M : {group gT}):
[/\ U \subset M, #|M : U| = p & extremal_class M = ModularGroup] ->
M :=: 'C_M('Mho^1(U)) /\ 'Ohm_1(M)%G \in 'E_p^2(M).
- case=> sUM iUM /modular_group_classP[q q_pr {n oU}[n n_gt23 isoM]].
have n_gt2: n > 2 by apply: leq_trans (leq_addl _ _) n_gt23.
have def_n: n = (n - 3).+3 by rewrite -{1}(subnKC n_gt2).
have oM: #|M| = (q ^ n)%N by rewrite (card_isog isoM) card_modular_group.
have pM: q.-group M by rewrite /pgroup oM pnatX pnat_id.
have def_q: q = p; last rewrite {q q_pr}def_q in oM pM isoM n_gt23.
by apply/eqP; rewrite eq_sym [p == q](p
...
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
cyclic_SCN
| |
normal_rank1_structuregT p (G : {group gT}) :
p.-group G -> (forall X : {group gT}, X <| G -> abelian X -> cyclic X) ->
cyclic G \/ [&& p == 2, extremal2 G & (#|G| >= 16) || (G \isog 'Q_8)].
Proof.
move=> pG dn_G_1.
have [cGG | not_cGG] := boolP (abelian G); first by left; rewrite dn_G_1.
have [X maxX]: {X | [max X | X <| G & abelian X]}.
by apply: ex_maxgroup; exists 1%G; rewrite normal1 abelian1.
have cycX: cyclic X by rewrite dn_G_1; case/andP: (maxgroupp maxX).
have scX: X \in 'SCN(G) := max_SCN pG maxX.
have [[p2 _ cG] | [M [_ _ _]]] := cyclic_SCN pG scX not_cGG cycX; last first.
rewrite 2!inE -andbA => /and3P[sEG abelE dimE_2] charE.
have:= dn_G_1 _ (char_normal charE) (abelem_abelian abelE).
by rewrite (abelem_cyclic abelE) (eqP dimE_2).
have [n oG] := p_natP pG; right; rewrite p2 cG /= in oG *.
rewrite oG (@leq_exp2l 2 4) //.
rewrite /extremal2 /extremal_class oG pfactorKpdiv // in cG.
case: andP cG => [[n_gt1 isoG] _ | _]; last first.
by case: (ltngtP 3 n) => //= <-; do 2?case: ifP.
have [[x y] genG _] := generators_2dihedral n_gt1 isoG.
have [_ _ _ [_ _ maxG]] := dihedral2_structure n_gt1 genG isoG.
rewrite 2!ltn_neqAle n_gt1 !(eq_sym _ n).
case: eqP => [_ abelG| _]; first by rewrite (abelem_abelian abelG) in not_cGG.
case: eqP => // -> [_ _ isoY _ _]; set Y := <<_>> in isoY.
have nxYG: Y <| G by rewrite (p_maximal_normal pG) // maxG !inE eqxx orbT.
have [// | [u v] genY _] := generators_2dihedral _ isoY.
case/dihedral2_structure: (genY) => //= _ _ _ _ abelY.
have:= dn_G_
...
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
normal_rank1_structure
| |
odd_pgroup_rank1_cyclicgT p (G : {group gT}) :
p.-group G -> odd #|G| -> cyclic G = ('r_p(G) <= 1).
Proof.
move=> pG oddG; rewrite -rank_pgroup //; apply/idP/idP=> [cycG | dimG1].
by rewrite -abelian_rank1_cyclic ?cyclic_abelian.
have [X nsXG cXX|//|] := normal_rank1_structure pG; last first.
by rewrite (negPf (odd_not_extremal2 oddG)) andbF.
by rewrite abelian_rank1_cyclic // (leq_trans (rankS (normal_sub nsXG))).
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
odd_pgroup_rank1_cyclic
| |
prime_Ohm1PgT p (G : {group gT}) :
p.-group G -> G :!=: 1 ->
reflect (#|'Ohm_1(G)| = p)
(cyclic G || (p == 2) && (extremal_class G == Quaternion)).
Proof.
move=> pG ntG; have [p_pr p_dvd_G _] := pgroup_pdiv pG ntG.
apply: (iffP idP) => [|oG1p].
case/orP=> [cycG|]; first exact: Ohm1_cyclic_pgroup_prime.
case/andP=> /eqP p2 /eqP/quaternion_classP[n n_gt2 isoG].
rewrite p2; have [[x y]] := generators_quaternion n_gt2 isoG.
by case/quaternion_structure=> // _ _ [<- oZ _ [->]].
have [X nsXG cXX|-> //|]:= normal_rank1_structure pG.
have [sXG _] := andP nsXG; have pX := pgroupS sXG pG.
rewrite abelian_rank1_cyclic // (rank_pgroup pX) p_rank_abelian //.
rewrite -{2}(pfactorK 1 p_pr) -{3}oG1p dvdn_leq_log ?cardG_gt0 //.
by rewrite cardSg ?OhmS.
case/and3P=> /eqP p2; rewrite p2 (orbC (cyclic G)) /extremal2.
case cG: (extremal_class G) => //; case: notF.
case/dihedral_classP: cG => n n_gt1 isoG.
have [[x y] genG _] := generators_2dihedral n_gt1 isoG.
have [oG _ _ _] := genG; case/dihedral2_structure: genG => // _ _ [defG1 _] _.
by case/idPn: n_gt1; rewrite -(@ltn_exp2l 2) // -oG -defG1 oG1p p2.
case/semidihedral_classP: cG => n n_gt3 isoG.
have [[x y] genG [oy _]] := generators_semidihedral n_gt3 isoG.
case/semidihedral_structure: genG => // _ _ [_ _ [defG1 _] _] _ [isoG1 _ _].
case/idPn: (n_gt3); rewrite -(ltn_predK n_gt3) ltnS -leqNgt -(@leq_exp2l 2) //.
rewrite -card_2dihedral //; last by rewrite -(subnKC n_gt3).
by rewrite -(card_isog isoG1) /= -defG1 oG
...
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
prime_Ohm1P
| |
symplectic_type_group_structuregT p (G : {group gT}) :
p.-group G -> (forall X : {group gT}, X \char G -> abelian X -> cyclic X) ->
exists2 E : {group gT}, E :=: 1 \/ extraspecial E
& exists R : {group gT},
[/\ cyclic R \/ [/\ p = 2, extremal2 R & #|R| >= 16],
E \* R = G
& E :&: R = 'Z(E)].
Proof.
move=> pG sympG; have [H [charH]] := Thompson_critical pG.
have sHG := char_sub charH; have pH := pgroupS sHG pG.
set U := 'Z(H) => sPhiH_U sHG_U defU; set Z := 'Ohm_1(U).
have sZU: Z \subset U by rewrite Ohm_sub.
have charU: U \char G := gFchar_trans _ charH.
have cUU: abelian U := center_abelian H.
have cycU: cyclic U by apply: sympG.
have pU: p.-group U := pgroupS (char_sub charU) pG.
have cHU: U \subset 'C(H) by rewrite subsetIr.
have cHsHs: abelian (H / Z).
rewrite sub_der1_abelian //= (OhmE _ pU) genS //= -/U.
apply/subsetP=> _ /imset2P[h k Hh Hk ->].
have Uhk: [~ h, k] \in U by rewrite (subsetP sHG_U) ?mem_commg ?(subsetP sHG).
rewrite inE Uhk inE -commXg; last by red; rewrite -(centsP cHU).
apply/commgP; red; rewrite (centsP cHU) // (subsetP sPhiH_U) //.
by rewrite (Phi_joing pH) mem_gen // inE orbC (Mho_p_elt 1) ?(mem_p_elt pH).
have nsZH: Z <| H by rewrite sub_center_normal.
have [K /=] := inv_quotientS nsZH (Ohm_sub 1 (H / Z)); fold Z => defKs sZK sKH.
have nsZK: Z <| K := normalS sZK sKH nsZH; have [_ nZK] := andP nsZK.
have abelKs: p.-abelem (K / Z) by rewrite -defKs Ohm1_abelem ?quotient_pgroup.
have charK: K \char G.
have charZ: Z \char H := gFchar_trans _ (c
...
|
Theorem
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotient action commutator gproduct gfunctor",
"From mathcomp Require Import ssralg countalg finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal"
] |
solvable/extremal.v
|
symplectic_type_group_structure
| |
fmod_of(gT : finGroupType) (A : {group gT}) (abelA : abelian A) :=
Fmod x & x \in A.
Bind Scope ring_scope with fmod_of.
|
Inductive
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
fmod_of
| |
fmvalu := val (f2sub_magma u).
#[export]
HB.instance Definition _ := [isSub for fmval].
Local Notation valA := (val: fmodA -> gT) (only parsing).
#[export]
HB.instance Definition _ := [Finite of fmodA by <:].
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
fmval
| |
fmodx := sub2f (subg A x).
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
fmod
| |
actru x := if x \in 'N(A) then fmod (fmval u ^ x) else u.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
actr
| |
fmod_oppu := sub2f u^-1.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
fmod_opp
| |
fmod_addu v := sub2f (u * v).
Fact fmod_add0r : left_id (sub2f 1) fmod_add.
Proof. by move=> u; apply: val_inj; apply: mul1g. Qed.
Fact fmod_addrA : associative fmod_add.
Proof. by move=> u v w; apply: val_inj; apply: mulgA. Qed.
Fact fmod_addNr : left_inverse (sub2f 1) fmod_opp fmod_add.
Proof. by move=> u; apply: val_inj; apply: mulVg. Qed.
Fact fmod_addrC : commutative fmod_add.
Proof. by case=> x Ax [y Ay]; apply: val_inj; apply: (centsP abelA). Qed.
#[export]
HB.instance Definition _ :=
GRing.isZmodule.Build fmodA fmod_addrA fmod_addrC fmod_add0r fmod_addNr.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
fmod_add
| |
Definition_ := [finGroupMixin of fmodA for +%R].
|
HB.instance
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
Definition
| |
fmodPu : val u \in A. Proof. exact: valP. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
fmodP
| |
fmod_inj: injective fmval. Proof. exact: val_inj. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
fmod_inj
| |
congr_fmodu v : u = v -> fmval u = fmval v.
Proof. exact: congr1. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
congr_fmod
| |
fmvalA: {morph valA : x y / x + y >-> (x * y)%g}. Proof. by []. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
fmvalA
| |
fmvalN: {morph valA : x / - x >-> x^-1%g}. Proof. by []. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
fmvalN
| |
fmval0: valA 0 = 1%g. Proof. by []. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
fmval0
| |
fmval_morphism:= @Morphism _ _ setT fmval (in2W fmvalA).
|
Canonical
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
fmval_morphism
| |
fmval_sum:= big_morph fmval fmvalA fmval0.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
fmval_sum
| |
fmvalZn : {morph valA : x / x *+ n >-> (x ^+ n)%g}.
Proof. by move=> u; rewrite /= morphX ?inE. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
fmvalZ
| |
fmodKcondx : val (fmod x) = if x \in A then x else 1%g.
Proof. by rewrite /= /fmval /= val_insubd. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
fmodKcond
| |
fmodK: {in A, cancel fmod val}. Proof. exact: subgK. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
fmodK
| |
fmvalK: cancel val fmod.
Proof. by case=> x Ax; apply: val_inj; rewrite /fmod /= sgvalK. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
fmvalK
| |
fmod1: fmod 1 = 0. Proof. by rewrite -fmval0 fmvalK. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
fmod1
| |
fmodM: {in A &, {morph fmod : x y / (x * y)%g >-> x + y}}.
Proof. by move=> x y Ax Ay /=; apply: val_inj; rewrite /fmod morphM. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
fmodM
| |
fmod_morphism:= Morphism fmodM.
|
Canonical
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
fmod_morphism
| |
fmodXn : {in A, {morph fmod : x / (x ^+ n)%g >-> x *+ n}}.
Proof. exact: morphX. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
fmodX
| |
fmodV: {morph fmod : x / x^-1%g >-> - x}.
Proof.
move=> x; apply: val_inj; rewrite fmvalN !fmodKcond groupV.
by case: (x \in A); rewrite ?invg1.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
fmodV
| |
injm_fmod: 'injm fmod.
Proof.
by apply/injmP=> x y Ax Ay []; move/val_inj; apply: (injmP (injm_subg A)).
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
injm_fmod
| |
fmvalJcondu x :
val (u ^@ x) = if x \in 'N(A) then val u ^ x else val u.
Proof. by case: ifP => Nx; rewrite /actr Nx ?fmodK // memJ_norm ?fmodP. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
fmvalJcond
| |
fmvalJu x : x \in 'N(A) -> val (u ^@ x) = val u ^ x.
Proof. by move=> Nx; rewrite fmvalJcond Nx. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
fmvalJ
| |
fmodJx y : y \in 'N(A) -> fmod (x ^ y) = fmod x ^@ y.
Proof.
move=> Ny; apply: val_inj; rewrite fmvalJ ?fmodKcond ?memJ_norm //.
by case: ifP => // _; rewrite conj1g.
Qed.
Fact actr_is_action : is_action 'N(A) actr.
Proof.
split=> [a u v eq_uv_a | u a b Na Nb].
case Na: (a \in 'N(A)); last by rewrite /actr Na in eq_uv_a.
by apply: val_inj; apply: (conjg_inj a); rewrite -!fmvalJ ?eq_uv_a.
by apply: val_inj; rewrite !fmvalJ ?groupM ?conjgM.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
fmodJ
| |
actr_action:= Action actr_is_action.
|
Canonical
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
actr_action
| |
act0rx : 0 ^@ x = 0.
Proof. by rewrite /actr conj1g morph1 if_same. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
act0r
| |
actArx : {morph actr^~ x : u v / u + v}.
Proof.
by move=> u v; apply: val_inj; rewrite !(fmvalA, fmvalJcond) conjMg; case: ifP.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
actAr
| |
actr_sumx := big_morph _ (actAr x) (act0r x).
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
actr_sum
| |
actNrx : {morph actr^~ x : u / - u}.
Proof. by move=> u; apply: (addrI (u ^@ x)); rewrite -actAr !subrr act0r. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
actNr
| |
actZrx n : {morph actr^~ x : u / u *+ n}.
Proof.
by move=> u; elim: n => [|n IHn]; rewrite ?act0r // !mulrS actAr IHn.
Qed.
Fact actr_is_groupAction : is_groupAction setT 'M.
Proof.
move=> a Na /[1!inE]; apply/andP; split; first by apply/subsetP=> u _ /[1!inE].
by apply/morphicP=> u v _ _; rewrite !permE /= actAr.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
actZr
| |
actr_groupAction:= GroupAction actr_is_groupAction.
|
Canonical
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
actr_groupAction
| |
actr1u : u ^@ 1 = u.
Proof. exact: act1. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
actr1
| |
actrM: {in 'N(A) &, forall x y u, u ^@ (x * y) = u ^@ x ^@ y}.
Proof.
by move=> x y Nx Ny /= u; apply: val_inj; rewrite !fmvalJ ?conjgM ?groupM.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
actrM
| |
actrKx : cancel (actr^~ x) (actr^~ x^-1%g).
Proof.
move=> u; apply: val_inj; rewrite !fmvalJcond groupV.
by case: ifP => -> //; rewrite conjgK.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
actrK
| |
actrKVx : cancel (actr^~ x^-1%g) (actr^~ x).
Proof. by move=> u; rewrite /= -{2}(invgK x) actrK. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
actrKV
| |
Gaschutz_split: [splits G, over H] = [splits P, over H].
Proof.
apply/splitsP/splitsP=> [[K /complP[tiHK eqHK]] | [Q /complP[tiHQ eqHQ]]].
exists (K :&: P)%G; rewrite inE setICA (setIidPl sHP) setIC tiHK eqxx.
by rewrite group_modl // eqHK (sameP eqP setIidPr).
have sQP: Q \subset P by rewrite -eqHQ mulG_subr.
pose rP x := repr (P :* x); pose pP x := x * (rP x)^-1.
have PpP x: pP x \in P by rewrite -mem_rcoset rcoset_repr rcoset_refl.
have rPmul x y: x \in P -> rP (x * y) = rP y.
by move=> Px; rewrite /rP rcosetM rcoset_id.
pose pQ x := remgr H Q x; pose rH x := pQ (pP x) * rP x.
have pQhq: {in H & Q, forall h q, pQ (h * q) = q} by apply: remgrMid.
have pQmul: {in P &, {morph pQ : x y / x * y}}.
by apply: remgrM; [apply/complP | apply: normalS (nsHG)].
have HrH x: rH x \in H :* x.
by rewrite rcoset_sym mem_rcoset invMg mulgA mem_divgr // eqHQ PpP.
have GrH x: x \in G -> rH x \in G.
move=> Gx; case/rcosetP: (HrH x) => y Hy ->.
by rewrite groupM // (subsetP sHG).
have rH_Pmul x y: x \in P -> rH (x * y) = pQ x * rH y.
by move=> Px; rewrite /rH mulgA -pQmul; first by rewrite /pP rPmul ?mulgA.
have rH_Hmul h y: h \in H -> rH (h * y) = rH y.
by move=> Hh; rewrite rH_Pmul ?(subsetP sHP) // -(mulg1 h) pQhq ?mul1g.
pose mu x y := fmod ((rH x * rH y)^-1 * rH (x * y)).
pose nu y := (\sum_(Px in rcosets P G) mu (repr Px) y)%R.
have rHmul: {in G &, forall x y, rH (x * y) = rH x * rH y * val (mu x y)}.
move=> x y Gx Gy; rewrite /= fmodK ?mulKVg // -mem_lcoset lcoset_sym.
rewrite -n
...
|
Theorem
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
Gaschutz_split
| |
Gaschutz_transitive: {in [complements to H in G] &,
forall K L, K :&: P = L :&: P -> exists2 x, x \in H & L :=: K :^ x}.
Proof.
move=> K L /=; set Q := K :&: P => /complP[tiHK eqHK] cpHL QeqLP.
have [trHL eqHL] := complP cpHL.
pose nu x := fmod (divgr H L x^-1).
have sKG: {subset K <= G} by apply/subsetP; rewrite -eqHK mulG_subr.
have sLG: {subset L <= G} by apply/subsetP; rewrite -eqHL mulG_subr.
have val_nu x: x \in G -> val (nu x) = divgr H L x^-1.
by move=> Gx; rewrite fmodK // mem_divgr // eqHL groupV.
have nu_cocycle: {in G &, forall x y, nu (x * y)%g = nu x ^@ y + nu y}%R.
move=> x y Gx Gy; apply: val_inj; rewrite fmvalA fmvalJ ?nHG //.
rewrite !val_nu ?groupM // /divgr conjgE !mulgA mulgK.
by rewrite !(invMg, remgrM cpHL) ?groupV ?mulgA.
have nuL x: x \in L -> nu x = 0%R.
move=> Lx; apply: val_inj; rewrite val_nu ?sLG //.
by rewrite /divgr remgr_id ?groupV ?mulgV.
exists (fmval ((\sum_(X in rcosets Q K) nu (repr X)) *+ m)).
exact: fmodP.
apply/eqP; rewrite eq_sym eqEcard; apply/andP; split; last first.
by rewrite cardJg -(leq_pmul2l (cardG_gt0 H)) -!TI_cardMg // eqHL eqHK.
apply/subsetP=> _ /imsetP[x Kx ->]; rewrite conjgE mulgA (conjgC _ x).
have Gx: x \in G by rewrite sKG.
rewrite conjVg -mulgA -fmvalJ ?nHG // -fmvalN -fmvalA (_ : _ + _ = nu x)%R.
by rewrite val_nu // mulKVg groupV mem_remgr // eqHL groupV.
rewrite actZr -!mulNrn -mulrnDl actr_sum.
rewrite addrC (reindex_acts _ (actsRs_rcosets _ K) Kx) -sumrB /= -/Q.
rewrite (eq_bigr (fun _ => nu x)) => [|_ /imset
...
|
Theorem
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
Gaschutz_transitive
| |
coprime_abel_cent_TI(gT : finGroupType) (A G : {group gT}) :
A \subset 'N(G) -> coprime #|G| #|A| -> abelian G -> 'C_[~: G, A](A) = 1.
Proof.
move=> nGA coGA abG; pose f x := val (\sum_(a in A) fmod abG x ^@ a)%R.
have fM: {in G &, {morph f : x y / x * y}}.
move=> x y Gx Gy /=; rewrite -fmvalA -big_split /=; congr (fmval _).
by apply: eq_bigr => a Aa; rewrite fmodM // actAr.
have nfA x a: a \in A -> f (x ^ a) = f x.
move=> Aa; rewrite {2}/f (reindex_inj (mulgI a)) /=; congr (fmval _).
apply: eq_big => [b | b Ab]; first by rewrite groupMl.
by rewrite -!fmodJ ?groupM ?(subsetP nGA) // conjgM.
have kerR: [~: G, A] \subset 'ker (Morphism fM).
rewrite gen_subG; apply/subsetP=> xa; case/imset2P=> x a Gx Aa -> {xa}.
have Gxa: x ^ a \in G by rewrite memJ_norm ?(subsetP nGA).
rewrite commgEl; apply/kerP; rewrite (groupM, morphM) ?(groupV, morphV) //=.
by rewrite nfA ?mulVg.
apply/trivgP; apply/subsetP=> x /setIP[Rx cAx]; apply/set1P.
have Gx: x \in G by apply: subsetP Rx; rewrite commg_subl.
rewrite -(expgK coGA Gx) (_ : x ^+ _ = 1) ?expg1n //.
rewrite -(fmodK abG Gx) -fmvalZ -(mker (subsetP kerR x Rx)); congr fmval.
rewrite -GRing.sumr_const; apply: eq_bigr => a Aa.
by rewrite -fmodJ ?(subsetP nGA) // /conjg (centP cAx) // mulKg.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
coprime_abel_cent_TI
| |
transferg := V repr g.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
transfer
| |
transferM: {in G &, {morph transfer : x y / (x * y)%g >-> x + y}}.
Proof.
move=> s t Gs Gt /=.
rewrite [transfer t](reindex_acts 'Rs _ Gs) ?actsRs_rcosets //= -big_split /=.
apply: eq_bigr => _ /rcosetsP[x Gx ->]; rewrite !rcosetE -!rcosetM.
rewrite -zmodMgE -morphM -?mem_rcoset; first by rewrite !mulgA mulgKV rcosetM.
by rewrite rcoset_repr rcosetM mem_rcoset mulgK mem_repr_rcoset.
by rewrite rcoset_repr (rcosetM _ _ t) mem_rcoset mulgK mem_repr_rcoset.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
transferM
| |
transfer_morphism:= Morphism transferM.
|
Canonical
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
transfer_morphism
| |
transfer_indepX (rX := transversal_repr 1 X) :
is_transversal X HG G -> {in G, transfer =1 V rX}.
Proof.
move=> trX g Gg; have mem_rX := repr_mem_pblock trX 1; rewrite -/rX in mem_rX.
apply: (addrI (\sum_(Hx in HG) fmalpha (repr Hx * (rX Hx)^-1))).
rewrite {1}(reindex_acts 'Rs _ Gg) ?actsRs_rcosets // -!big_split /=.
apply: eq_bigr => _ /rcosetsP[x Gx ->]; rewrite !rcosetE -!rcosetM.
case: repr_rcosetP => h1 Hh1; case: repr_rcosetP => h2 Hh2.
have: H :* (x * g) \in rcosets H G by rewrite -rcosetE imset_f ?groupM.
have: H :* x \in rcosets H G by rewrite -rcosetE imset_f.
case/mem_rX/rcosetP=> h3 Hh3 -> /mem_rX/rcosetP[h4 Hh4 ->].
rewrite -!(mulgA h1) -!(mulgA h2) -!(mulgA h3).
do 3 rewrite invMg mulKVg.
by rewrite addrC -!zmodMgE -!morphM ?groupM ?groupV // -!mulgA !mulKg.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
transfer_indep
| |
mulg_exp_card_rcosetsx : x * (g ^+ n_ x) \in H :* x.
Proof.
rewrite /n_ /indexg -orbitRs -porbit_actperm ?inE //.
rewrite -{2}(iter_porbit (actperm 'Rs g) (H :* x)) -permX -morphX ?inE //.
by rewrite actpermE //= rcosetE -rcosetM rcoset_refl.
Qed.
Let HGg : {set {set {set gT}}} := orbit 'Rs <[g]> @: HG.
Let partHG : partition HG G := rcosets_partition sHG.
Let actsgHG : [acts <[g]>, on HG | 'Rs].
Proof. exact: subset_trans sgG (actsRs_rcosets H G). Qed.
Let partHGg : partition HGg HG := orbit_partition actsgHG.
Let injHGg : {in HGg &, injective cover}.
Proof. by have [] := partition_partition partHG partHGg. Qed.
Let defHGg : HG :* <[g]> = cover @: HGg.
Proof.
rewrite -imset_comp [_ :* _]imset2_set1r; apply: eq_imset => Hx /=.
by rewrite cover_imset -curry_imset2r.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
mulg_exp_card_rcosets
| |
rcosets_cycle_partition: partition (HG :* <[g]>) G.
Proof. by rewrite defHGg; have [] := partition_partition partHG partHGg. Qed.
Variable X : {set gT}.
Hypothesis trX : is_transversal X (HG :* <[g]>) G.
Let sXG : {subset X <= G}. Proof. exact/subsetP/(transversal_sub trX). Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
rcosets_cycle_partition
| |
rcosets_cycle_transversal: H_g_rcosets @: X = HGg.
Proof.
have sHXgHGg x: x \in X -> H_g_rcosets x \in HGg.
by move/sXG=> Gx; apply: imset_f; rewrite -rcosetE imset_f.
apply/setP=> Hxg; apply/imsetP/idP=> [[x /sHXgHGg HGgHxg -> //] | HGgHxg].
have [_ /rcosetsP[z Gz ->] ->] := imsetP HGgHxg.
pose Hzg := H :* z * <[g]>; pose x := transversal_repr 1 X Hzg.
have HGgHzg: Hzg \in HG :* <[g]>.
by rewrite mem_mulg ?set11 // -rcosetE imset_f.
have Hzg_x: x \in Hzg by rewrite (repr_mem_pblock trX).
exists x; first by rewrite (repr_mem_transversal trX).
case/mulsgP: Hzg_x => y u /rcoset_eqP <- /(orbit_act 'Rs) <- -> /=.
by rewrite rcosetE -rcosetM.
Qed.
Local Notation defHgX := rcosets_cycle_transversal.
Let injHg: {in X &, injective H_g_rcosets}.
Proof.
apply/imset_injP; rewrite defHgX (card_transversal trX) defHGg.
by rewrite (card_in_imset injHGg).
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
rcosets_cycle_transversal
| |
sum_index_rcosets_cycle: (\sum_(x in X) n_ x)%N = #|G : H|.
Proof. by rewrite [#|G : H|](card_partition partHGg) -defHgX big_imset. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
sum_index_rcosets_cycle
| |
transfer_cycle_expansion:
transfer g = \sum_(x in X) fmalpha ((g ^+ n_ x) ^ (x^-1)).
Proof.
pose Y := \bigcup_(x in X) [set x * g ^+ i | i : 'I_(n_ x)].
pose rY := transversal_repr 1 Y.
pose pcyc x := porbit (actperm 'Rs g) (H :* x).
pose traj x := traject (actperm 'Rs g) (H :* x) #|pcyc x|.
have Hgr_eq x: H_g_rcosets x = pcyc x.
by rewrite /H_g_rcosets -orbitRs -porbit_actperm ?inE.
have pcyc_eq x: pcyc x =i traj x by apply: porbit_traject.
have uniq_traj x: uniq (traj x) by apply: uniq_traject_porbit.
have n_eq x: n_ x = #|pcyc x| by rewrite -Hgr_eq.
have size_traj x: size (traj x) = n_ x by rewrite n_eq size_traject.
have nth_traj x j: j < n_ x -> nth (H :* x) (traj x) j = H :* (x * g ^+ j).
move=> lt_j_x; rewrite nth_traject -?n_eq //.
by rewrite -permX -morphX ?inE // actpermE //= rcosetE rcosetM.
have sYG: Y \subset G.
apply/bigcupsP=> x Xx; apply/subsetP=> _ /imsetP[i _ ->].
by rewrite groupM ?groupX // sXG.
have trY: is_transversal Y HG G.
apply/and3P; split=> //; apply/forall_inP=> Hy.
have /and3P[/eqP <- _ _] := partHGg; rewrite -defHgX cover_imset.
case/bigcupP=> x Xx; rewrite Hgr_eq pcyc_eq => /trajectP[i].
rewrite -n_eq -permX -morphX ?in_setT // actpermE /= rcosetE -rcosetM => lti.
set y := x * _ => ->{Hy}; pose oi := Ordinal lti.
have Yy: y \in Y by apply/bigcupP; exists x => //; apply/imsetP; exists oi.
apply/cards1P; exists y; apply/esym/eqP.
rewrite eqEsubset sub1set inE Yy rcoset_refl.
apply/subsetP=> _ /setIP[/bigcupP[x' Xx' /imsetP[j _ ->]] Hy_x'gj
...
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import cyclic"
] |
solvable/finmodule.v
|
transfer_cycle_expansion
| |
semiregularK H := {in H^#, forall x, 'C_K[x] = 1}.
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
semiregular
| |
semiprimeK H := {in H^#, forall x, 'C_K[x] = 'C_K(H)}.
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
semiprime
| |
normedTIA G L := [&& A != set0, trivIset (A :^: G) & 'N_G(A) == L].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
normedTI
| |
Frobenius_group_with_complementG H := (H != G) && normedTI H^# G H.
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
Frobenius_group_with_complement
| |
Frobenius_groupG :=
[exists H : {group gT}, Frobenius_group_with_complement G H].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
Frobenius_group
| |
Frobenius_group_with_kernel_and_complementG K H :=
(K ><| H == G) && Frobenius_group_with_complement G H.
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
Frobenius_group_with_kernel_and_complement
| |
Frobenius_group_with_kernelG K :=
[exists H : {group gT}, Frobenius_group_with_kernel_and_complement G K H].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] |
solvable/frobenius.v
|
Frobenius_group_with_kernel
|
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