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 |
|---|---|---|---|---|---|---|
cyclicJG x : cyclic (G :^ x) = cyclic G.
Proof.
apply/cyclicP/cyclicP=> [[y /(canRL (conjsgK x))] | [y ->]].
by rewrite -cycleJ; exists (y ^ x^-1).
by exists (y ^ x); rewrite cycleJ.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
cyclicJ
| |
eq_subG_cyclicG H K :
cyclic G -> H \subset G -> K \subset G -> (H :==: K) = (#|H| == #|K|).
Proof.
case/cyclicP=> x -> sHx sKx; apply/eqP/eqP=> [-> //| eqHK].
have def_GHx := cycle_sub_group (cardSg sHx); set GHx := [set _] in def_GHx.
have []: H \in GHx /\ K \in GHx by rewrite -def_GHx !inE sHx sKx eqHK /=.
by do 2!move/set1P->.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
eq_subG_cyclic
| |
cardSg_cyclicG H K :
cyclic G -> H \subset G -> K \subset G -> (#|H| %| #|K|) = (H \subset K).
Proof.
move=> cycG sHG sKG; apply/idP/idP; last exact: cardSg.
case/cyclicP: (cyclicS sKG cycG) => x defK; rewrite {K}defK in sKG *.
case/dvdnP=> k ox; suffices ->: H :=: <[x ^+ k]> by apply: cycleX.
apply/eqP; rewrite (eq_subG_cyclic cycG) ?(subset_trans (cycleX _ _)) //.
rewrite -orderE orderXdiv orderE ox ?dvdn_mulr ?mulKn //.
by have:= order_gt0 x; rewrite orderE ox; case k.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
cardSg_cyclic
| |
sub_cyclic_charG H : cyclic G -> (H \char G) = (H \subset G).
Proof.
case/cyclicP=> x ->; apply/idP/idP => [/andP[] //|].
exact: cycle_subgroup_char.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
sub_cyclic_char
| |
morphim_cyclicrT G H (f : {morphism G >-> rT}) :
cyclic H -> cyclic (f @* H).
Proof.
move=> cycH; wlog sHG: H cycH / H \subset G.
by rewrite -morphimIdom; apply; rewrite (cyclicS _ cycH, subsetIl) ?subsetIr.
case/cyclicP: cycH sHG => x ->; rewrite gen_subG sub1set => Gx.
by apply/cyclicP; exists (f x); rewrite morphim_cycle.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
morphim_cyclic
| |
quotient_cyclex H : x \in 'N(H) -> <[x]> / H = <[coset H x]>.
Proof. exact: morphim_cycle. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
quotient_cycle
| |
quotient_cyclicG H : cyclic G -> cyclic (G / H).
Proof. exact: morphim_cyclic. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
quotient_cyclic
| |
quotient_generatorx G H :
x \in 'N(H) -> generator G x -> generator (G / H) (coset H x).
Proof. by move=> Nx; apply: morph_generator. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
quotient_generator
| |
prime_cyclicG : prime #|G| -> cyclic G.
Proof.
case/primeP; rewrite ltnNge -trivg_card_le1.
case/trivgPn=> x Gx ntx /(_ _ (order_dvdG Gx)).
rewrite order_eq1 (negbTE ntx) => /eqnP oxG; apply/cyclicP.
by exists x; apply/eqP; rewrite eq_sym eqEcard -oxG cycle_subG Gx leqnn.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
prime_cyclic
| |
dvdn_prime_cyclicG p : prime p -> #|G| %| p -> cyclic G.
Proof.
move=> p_pr pG; case: (eqsVneq G 1) => [-> | ntG]; first exact: cyclic1.
by rewrite prime_cyclic // (prime_nt_dvdP p_pr _ pG) -?trivg_card1.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
dvdn_prime_cyclic
| |
cyclic_smallG : #|G| <= 3 -> cyclic G.
Proof.
rewrite 4!(ltnS, leq_eqVlt) -trivg_card_le1 orbA orbC.
case/predU1P=> [-> | oG]; first exact: cyclic1.
by apply: prime_cyclic; case/pred2P: oG => ->.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
cyclic_small
| |
injm_cyclicG H (f : {morphism G >-> rT}) :
'injm f -> H \subset G -> cyclic (f @* H) = cyclic H.
Proof.
move=> injf sHG; apply/idP/idP; last exact: morphim_cyclic.
by rewrite -{2}(morphim_invm injf sHG); apply: morphim_cyclic.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
injm_cyclic
| |
isog_cyclicG M : G \isog M -> cyclic G = cyclic M.
Proof. by case/isogP=> f injf <-; rewrite injm_cyclic. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
isog_cyclic
| |
isog_cyclic_cardG M : cyclic G -> isog G M = cyclic M && (#|M| == #|G|).
Proof.
move=> cycG; apply/idP/idP=> [isoGM | ].
by rewrite (card_isog isoGM) -(isog_cyclic isoGM) cycG /=.
case/cyclicP: cycG => x ->{G} /andP[/cyclicP[y ->] /eqP oy].
by apply: isog_trans (isog_symr _) (Zp_isog y); rewrite /order oy Zp_isog.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
isog_cyclic_card
| |
injm_generatorG H (f : {morphism G >-> rT}) x :
'injm f -> x \in G -> H \subset G ->
generator (f @* H) (f x) = generator H x.
Proof.
move=> injf Gx sHG; apply/idP/idP; last exact: morph_generator.
rewrite -{2}(morphim_invm injf sHG) -{2}(invmE injf Gx).
by apply: morph_generator; apply: mem_morphim.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
injm_generator
| |
metacyclicA :=
[exists H : {group gT}, [&& cyclic H, H <| A & cyclic (A / H)]].
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
metacyclic
| |
metacyclicPA :
reflect (exists H : {group gT}, [/\ cyclic H, H <| A & cyclic (A / H)])
(metacyclic A).
Proof. exact: 'exists_and3P. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
metacyclicP
| |
metacyclic1: metacyclic 1.
Proof.
by apply/existsP; exists 1%G; rewrite normal1 trivg_quotient !cyclic1.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
metacyclic1
| |
cyclic_metacyclicA : cyclic A -> metacyclic A.
Proof.
case/cyclicP=> x ->; apply/existsP; exists (<[x]>)%G.
by rewrite normal_refl cycle_cyclic trivg_quotient cyclic1.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
cyclic_metacyclic
| |
metacyclicSG H : H \subset G -> metacyclic G -> metacyclic H.
Proof.
move=> sHG /metacyclicP[K [cycK nsKG cycGq]]; apply/metacyclicP.
exists (H :&: K)%G; rewrite (cyclicS (subsetIr H K)) ?(normalGI sHG) //=.
rewrite setIC (isog_cyclic (second_isog _)) ?(cyclicS _ cycGq) ?quotientS //.
by rewrite (subset_trans sHG) ?normal_norm.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
metacyclicS
| |
cyclemof gT := fun x : gT => x ^+ n.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
cyclem
| |
cyclemM: {in <[a]> & , {morph cyclem a : x y / x * y}}.
Proof.
by move=> x y ax ay; apply: expgMn; apply: (centsP (cycle_abelian a)).
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
cyclemM
| |
cyclem_morphism:= Morphism cyclemM.
|
Canonical
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
cyclem_morphism
| |
injm_cyclem: 'injm (cyclem (val u) a).
Proof.
apply/subsetP=> x /setIdP[ax]; rewrite !inE -order_dvdn.
have [a1 | nta] := eqVneq a 1; first by rewrite a1 cycle1 inE in ax.
rewrite -order_eq1 -dvdn1; move/eqnP: (valP u) => /= <-.
by rewrite dvdn_gcd [in X in X && _]Zp_cast ?order_gt1 // order_dvdG.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
injm_cyclem
| |
im_cyclem: cyclem (val u) a @* <[a]> = <[a]>.
Proof.
apply/morphim_fixP=> //; first exact: injm_cyclem.
by rewrite morphim_cycle ?cycle_id ?cycleX.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
im_cyclem
| |
Zp_unitm:= aut injm_cyclem im_cyclem.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
Zp_unitm
| |
Zp_unitmM: {in units_Zp #[a] &, {morph Zp_unitm : u v / u * v}}.
Proof.
move=> u v _ _; apply: (eq_Aut (Aut_aut _ _)) => [|x a_x].
by rewrite groupM ?Aut_aut.
rewrite permM !autE ?groupX //= /cyclem -expgM.
rewrite -expg_mod_order modn_dvdm ?expg_mod_order //.
case: (leqP #[a] 1) => [lea1 | lt1a]; last by rewrite Zp_cast ?order_dvdG.
by rewrite card_le1_trivg // in a_x; rewrite (set1P a_x) order1 dvd1n.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
Zp_unitmM
| |
Zp_unit_morphism:= Morphism Zp_unitmM.
|
Canonical
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
Zp_unit_morphism
| |
injm_Zp_unitm: 'injm Zp_unitm.
Proof.
have [a1 | nta] := eqVneq a 1.
by rewrite subIset //= card_le1_trivg ?subxx // card_units_Zp a1 order1.
apply/subsetP=> /= u /morphpreP[_ /set1P/= um1].
have{um1}: Zp_unitm u a == Zp_unitm 1 a by rewrite um1 morph1.
rewrite !autE ?cycle_id // eq_expg_mod_order.
by rewrite -[n in _ == _ %[mod n]]Zp_cast ?order_gt1 // !modZp inE.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
injm_Zp_unitm
| |
generator_coprimem : generator <[a]> (a ^+ m) = coprime #[a] m.
Proof.
rewrite /generator eq_sym eqEcard cycleX -/#[a] [#|_|]orderXgcd /=.
apply/idP/idP=> [le_a_am|co_am]; last by rewrite (eqnP co_am) divn1.
have am_gt0: 0 < gcdn #[a] m by rewrite gcdn_gt0 order_gt0.
by rewrite /coprime eqn_leq am_gt0 andbT -(@leq_pmul2l #[a]) ?muln1 -?leq_divRL.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
generator_coprime
| |
im_Zp_unitm: Zp_unitm @* units_Zp #[a] = Aut <[a]>.
Proof.
rewrite morphimEdom; apply/setP=> f; pose n := invm (injm_Zpm a) (f a).
apply/imsetP/idP=> [[u _ ->] | Af]; first exact: Aut_aut.
have [a1 | nta] := eqVneq a 1.
by rewrite a1 cycle1 Aut1 in Af; exists 1; rewrite // morph1 (set1P Af).
have a_fa: <[a]> = <[f a]>.
by rewrite -(autmE Af) -morphim_cycle ?im_autm ?cycle_id.
have def_n: a ^+ n = f a.
by rewrite -/(Zpm n) invmK // im_Zpm a_fa cycle_id.
have co_a_n: coprime #[a].-2.+2 n.
by rewrite {1}Zp_cast ?order_gt1 // -generator_coprime def_n; apply/eqP.
exists (FinRing.unit 'Z_#[a] co_a_n); rewrite ?inE //.
apply: eq_Aut (Af) (Aut_aut _ _) _ => x ax.
rewrite autE //= /cyclem; case/cycleP: ax => k ->{x}.
by rewrite -(autmE Af) morphX ?cycle_id //= autmE -def_n -!expgM mulnC.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
im_Zp_unitm
| |
Zp_unit_isom: isom (units_Zp #[a]) (Aut <[a]>) Zp_unitm.
Proof. by apply/isomP; rewrite ?injm_Zp_unitm ?im_Zp_unitm. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
Zp_unit_isom
| |
Zp_unit_isog: isog (units_Zp #[a]) (Aut <[a]>).
Proof. exact: isom_isog Zp_unit_isom. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
Zp_unit_isog
| |
card_Aut_cycle: #|Aut <[a]>| = totient #[a].
Proof. by rewrite -(card_isog Zp_unit_isog) card_units_Zp. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
card_Aut_cycle
| |
totient_gen: totient #[a] = #|[set x | generator <[a]> x]|.
Proof.
have [lea1 | lt1a] := leqP #[a] 1.
rewrite /order card_le1_trivg // cards1 (@eq_card1 _ 1) // => x.
by rewrite !inE -cycle_eq1 eq_sym.
rewrite -(card_injm (injm_invm (injm_Zpm a))) /= ?im_Zpm; last first.
by apply/subsetP=> x /[1!inE]; apply: cycle_generator.
rewrite -card_units_Zp // cardsE card_sub morphim_invmE; apply: eq_card => /= d.
by rewrite !inE /= qualifE /= /Zp lt1a inE /= generator_coprime {1}Zp_cast.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
totient_gen
| |
Aut_cycle_abelian: abelian (Aut <[a]>).
Proof. by rewrite -im_Zp_unitm morphim_abelian ?units_Zp_abelian. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
Aut_cycle_abelian
| |
Aut_cyclic_abelian: cyclic G -> abelian (Aut G).
Proof. by case/cyclicP=> x ->; apply: Aut_cycle_abelian. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
Aut_cyclic_abelian
| |
card_Aut_cyclic: cyclic G -> #|Aut G| = totient #|G|.
Proof. by case/cyclicP=> x ->; apply: card_Aut_cycle. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
card_Aut_cyclic
| |
sum_ncycle_totient:
\sum_(d < #|G|.+1) #|[set <[x]> | x in G & #[x] == d]| * totient d = #|G|.
Proof.
pose h (x : gT) : 'I_#|G|.+1 := inord #[x].
symmetry; rewrite -{1}sum1_card (partition_big h xpredT) //=.
apply: eq_bigr => d _; set Gd := finset _.
rewrite -sum_nat_const sum1dep_card -sum1_card (_ : finset _ = Gd); last first.
apply/setP=> x /[!inE]; apply: andb_id2l => Gx.
by rewrite /eq_op /= inordK // ltnS subset_leq_card ?cycle_subG.
rewrite (partition_big_imset cycle) {}/Gd; apply: eq_bigr => C /=.
case/imsetP=> x /setIdP[Gx /eqP <-] -> {C d}.
rewrite sum1dep_card totient_gen; apply: eq_card => y; rewrite !inE /generator.
move: Gx; rewrite andbC eq_sym -!cycle_subG /order.
by case: eqP => // -> ->; rewrite eqxx.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
sum_ncycle_totient
| |
sum_totient_dvdn : \sum_(d < n.+1 | d %| n) totient d = n.
Proof.
case: n => [|[|n']]; try by rewrite big_mkcond !big_ord_recl big_ord0.
set n := n'.+2; pose x1 : 'Z_n := 1%R.
have ox1: #[x1] = n by rewrite /order -Zp_cycle card_Zp.
rewrite -[rhs in _ = rhs]ox1 -[#[_]]sum_ncycle_totient [#|_|]ox1 big_mkcond /=.
apply: eq_bigr => d _; rewrite -{2}ox1; case: ifP => [|ndv_dG]; last first.
rewrite eq_card0 // => C; apply/imsetP=> [[x /setIdP[Gx oxd] _{C}]].
by rewrite -(eqP oxd) order_dvdG in ndv_dG.
move/cycle_sub_group; set Gd := [set _] => def_Gd.
rewrite (_ : _ @: _ = @gval _ @: Gd); first by rewrite imset_set1 cards1 mul1n.
apply/setP=> C; apply/idP/imsetP=> [| [gC GdC ->{C}]].
case/imsetP=> x /setIdP[_ oxd] ->; exists <[x]>%G => //.
by rewrite -def_Gd inE -Zp_cycle subsetT.
have:= GdC; rewrite -def_Gd => /setIdP[_ /eqP <-].
by rewrite (set1P GdC) /= imset_f // inE eqxx (mem_cycle x1).
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
sum_totient_dvd
| |
order_inj_cyclic:
{in G &, forall x y, #[x] = #[y] -> <[x]> = <[y]>} -> cyclic G.
Proof.
move=> ucG; apply: negbNE (contra _ (negbT (ltnn #|G|))) => ncG.
rewrite -{2}[#|G|]sum_totient_dvd big_mkcond (bigD1 ord_max) ?dvdnn //=.
rewrite -{1}[#|G|]sum_ncycle_totient (bigD1 ord_max) //= -addSn leq_add //.
rewrite eq_card0 ?totient_gt0 ?cardG_gt0 // => C.
apply/imsetP=> [[x /setIdP[Gx /eqP oxG]]]; case/cyclicP: ncG.
by exists x; apply/eqP; rewrite eq_sym eqEcard cycle_subG Gx -oxG /=.
elim/big_ind2: _ => // [m1 n1 m2 n2 | d _]; first exact: leq_add.
set Gd := _ @: _; case: (set_0Vmem Gd) => [-> | [C]]; first by rewrite cards0.
rewrite {}/Gd => /imsetP[x /setIdP[Gx /eqP <-] _ {C d}].
rewrite order_dvdG // (@eq_card1 _ <[x]>) ?mul1n // => C.
apply/idP/eqP=> [|-> {C}]; last by rewrite imset_f // inE Gx eqxx.
by case/imsetP=> y /setIdP[Gy /eqP/ucG->].
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
order_inj_cyclic
| |
div_ring_mul_group_cyclic(R : unitRingType) (f : gT -> R) :
f 1 = 1%R -> {in G &, {morph f : u v / u * v >-> (u * v)%R}} ->
{in G^#, forall x, f x - 1 \in GRing.unit}%R ->
abelian G -> cyclic G.
Proof.
move=> f1 fM f1P abelG.
have fX n: {in G, {morph f : u / u ^+ n >-> (u ^+ n)%R}}.
by case: n => // n x Gx; elim: n => //= n IHn; rewrite expgS fM ?groupX ?IHn.
have fU x: x \in G -> f x \in GRing.unit.
by move=> Gx; apply/unitrP; exists (f x^-1); rewrite -!fM ?groupV ?gsimp.
apply: order_inj_cyclic => x y Gx Gy; set n := #[x] => yn.
apply/eqP; rewrite eq_sym eqEcard -[#|_|]/n yn leqnn andbT cycle_subG /=.
suff{y Gy yn} ->: <[x]> = G :&: [set z | #[z] %| n] by rewrite !inE Gy yn /=.
apply/eqP; rewrite eqEcard subsetI cycle_subG {}Gx /= cardE; set rs := enum _.
apply/andP; split; first by apply/subsetP=> y xy; rewrite inE order_dvdG.
pose P : {poly R} := ('X^n - 1)%R; have n_gt0: n > 0 by apply: order_gt0.
have szP : size P = n.+1.
by rewrite size_polyDl size_polyXn ?size_polyN ?size_poly1.
rewrite -ltnS -szP -(size_map f) max_ring_poly_roots -?size_poly_eq0 ?{}szP //.
apply/allP=> fy /mapP[y]; rewrite mem_enum !inE order_dvdn => /andP[Gy].
move/eqP=> yn1 ->{fy}; apply/eqP.
by rewrite !(hornerE, hornerXn) -fX // yn1 f1 subrr.
have: uniq rs by apply: enum_uniq.
have: all [in G] rs by apply/allP=> y; rewrite mem_enum; case/setIP.
elim: rs => //= y rs IHrs /andP[Gy Grs] /andP[y_rs]; rewrite andbC.
move/IHrs=> -> {IHrs}//; apply/allP=> _ /mapP[z rs_z ->].
have{Grs} Gz := allP Grs z rs_z
...
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
div_ring_mul_group_cyclic
| |
field_mul_group_cyclic(F : fieldType) (f : gT -> F) :
{in G &, {morph f : u v / u * v >-> (u * v)%R}} ->
{in G, forall x, f x = 1%R <-> x = 1} ->
cyclic G.
Proof.
move=> fM f1P; have f1 : f 1 = 1%R by apply/f1P.
apply: (div_ring_mul_group_cyclic f1 fM) => [x|].
case/setD1P=> x1 Gx; rewrite unitfE; apply: contra x1.
by rewrite subr_eq0 => /eqP/f1P->.
apply/centsP=> x Gx y Gy; apply/commgP/eqP.
apply/f1P; rewrite ?fM ?groupM ?groupV //.
by rewrite mulrCA -!fM ?groupM ?groupV // mulKg mulVg.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
field_mul_group_cyclic
| |
field_unit_group_cyclic(F : finFieldType) (G : {group {unit F}}) :
cyclic G.
Proof.
apply: field_mul_group_cyclic FinRing.uval _ _ => // u _.
by split=> /eqP ?; apply/eqP.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
field_unit_group_cyclic
| |
units_Zp_cyclicp : prime p -> cyclic (units_Zp p).
Proof. by move/pdiv_id <-; exact: field_unit_group_cyclic. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
units_Zp_cyclic
| |
has_prim_root_subproof(F : fieldType) (n : nat) (rs : seq F)
(n_gt0 : n > 0)
(rsn1 : all n.-unity_root rs)
(Urs : uniq rs)
(sz_rs : size rs = n)
(r := fun s => val (s : seq_sub rs))
(rn1 : forall x : seq_sub rs, r x ^+ n = 1)
(prim_r : forall z : F, z ^+ n = 1 -> z \in rs)
(r' := (fun s (e : s ^+ n = 1) => {| ssval := s; ssvalP := prim_r s e |})
: forall s : F, s ^+ n = 1 -> seq_sub rs)
(sG_1 := r' 1 (expr1n F n) : seq_sub rs)
(sG_VP : forall s : seq_sub rs, r s ^+ n.-1 ^+ n = 1)
(sG_MP : forall s s0 : seq_sub rs, (r s * r s0) ^+ n = 1)
(sG_V := (fun s : seq_sub rs => r' (r s ^+ n.-1) (sG_VP s))
: seq_sub rs -> seq_sub rs)
(sG_M := (fun s s0 : seq_sub rs => r' (r s * r s0) (sG_MP s s0))
: seq_sub rs -> seq_sub rs -> seq_sub rs)
(sG_Ag : associative sG_M)
(sG_1g : left_id sG_1 sG_M)
(sG_Vg : left_inverse sG_1 sG_V sG_M) :
has n.-primitive_root rs.
Proof.
pose ssMG : Finite_isGroup (seq_sub rs) := Finite_isGroup.Build (seq_sub rs) sG_Ag sG_1g sG_Vg.
pose gT : finGroupType := HB.pack (seq_sub rs) ssMG.
have /cyclicP[x gen_x]: @cyclic gT setT.
apply: (@field_mul_group_cyclic gT [set: _] F r) => // x _.
by split=> [ri1 | ->]; first apply: val_inj.
apply/hasP; exists (r x); first exact: (valP x).
have [m prim_x dvdmn] := prim_order_exists n_gt0 (rn1 x).
rewrite -((m =P n) _) // eqn_dvd {}dvdmn -sz_rs -(card_seq_sub Urs) -cardsT.
rewrite gen_x (@order_dvdn gT) /(_ == _) /= -{prim_x}(prim_expr_order prim_x).
by appl
...
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
has_prim_root_subproof
| |
has_prim_root(F : fieldType) (n : nat) (rs : seq F) :
n > 0 -> all n.-unity_root rs -> uniq rs -> size rs >= n ->
has n.-primitive_root rs.
Proof.
move=> n_gt0 rsn1 Urs; rewrite leq_eqVlt ltnNge max_unity_roots // orbF eq_sym.
move/eqP=> sz_rs; pose r := val (_ : seq_sub rs).
have rn1 x: r x ^+ n = 1.
by apply/eqP; rewrite -unity_rootE (allP rsn1) ?(valP x).
have prim_r z: z ^+ n = 1 -> z \in rs.
by move/eqP; rewrite -unity_rootE -(mem_unity_roots n_gt0).
pose r' := SeqSub (prim_r _ _); pose sG_1 := r' _ (expr1n _ _).
have sG_VP: r _ ^+ n.-1 ^+ n = 1.
by move=> x; rewrite -exprM mulnC exprM rn1 expr1n.
have sG_MP: (r _ * r _) ^+ n = 1 by move=> x y; rewrite exprMn !rn1 mul1r.
pose sG_V := r' _ (sG_VP _); pose sG_M := r' _ (sG_MP _ _).
have sG_Ag: associative sG_M by move=> x y z; apply: val_inj; rewrite /= mulrA.
have sG_1g: left_id sG_1 sG_M by move=> x; apply: val_inj; rewrite /= mul1r.
have sG_Vg: left_inverse sG_1 sG_V sG_M.
by move=> x; apply: val_inj; rewrite /= -exprSr prednK ?rn1.
exact: has_prim_root_subproof.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
has_prim_root
| |
Aut_prime_cycle_cyclic(a : gT) : prime #[a] -> cyclic (Aut <[a]>).
Proof.
move=> pr_a; have inj_um := injm_Zp_unitm a.
have /eq_S/eq_S eq_a := Fp_Zcast pr_a.
pose fm := cast_ord (esym eq_a) \o val \o invm inj_um.
apply: (@field_mul_group_cyclic _ _ _ fm) => [f g Af Ag | f Af] /=.
by apply: val_inj; rewrite /= morphM ?im_Zp_unitm //= eq_a.
split=> [/= fm1 |->]; last by apply: val_inj; rewrite /= morph1.
apply: (injm1 (injm_invm inj_um)); first by rewrite /= im_Zp_unitm.
by do 2!apply: val_inj; move/(congr1 val): fm1.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
Aut_prime_cycle_cyclic
| |
Aut_prime_cyclic(G : {group gT}) : prime #|G| -> cyclic (Aut G).
Proof.
move=> pr_G; case/cyclicP: (prime_cyclic pr_G) (pr_G) => x ->.
exact: Aut_prime_cycle_cyclic.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm automorphism quotient gproduct ssralg",
"From mathcomp Require Import finalg zmodp poly"
] |
solvable/cyclic.v
|
Aut_prime_cyclic
| |
actij (k : 'Z_p) := let: (i, j) := ij in (i + k * j, j)%R.
|
Definition
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
act
| |
actP: is_action [set: 'Z_p] act.
Proof.
apply: is_total_action=> [] [i j] => [|k1 k2] /=; first by rewrite mul0r addr0.
by rewrite mulrDl addrA.
Qed.
|
Lemma
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
actP
| |
action:= Action actP.
|
Canonical
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
action
| |
gactP: is_groupAction [set: 'Z_p * 'Z_p] action.
Proof.
move=> k _ /[1!inE]; apply/andP; split; first by apply/subsetP=> ij _ /[1!inE].
apply/morphicP=> /= [[i1 j1] [i2 j2] _ _].
by rewrite !permE /= mulrDr -addrA (addrCA i2) (addrA i1).
Qed.
|
Lemma
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
gactP
| |
groupAction:= GroupAction gactP.
Fact gtype_key : unit. Proof. by []. Qed.
|
Definition
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
groupAction
| |
gtype:= locked_with gtype_key (sdprod_groupType groupAction).
|
Definition
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
gtype
| |
ngtype:= ncprod [set: gtype].
|
Definition
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
ngtype
| |
ngtypeQn := xcprod [set: ngtype 2 n] 'Q_8.
|
Definition
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
ngtypeQ
| |
card_pX1p2: #|p^{1+2}| = (p ^ 3)%N.
Proof.
rewrite [@gtype _]unlock -(sdprod_card (sdprod_sdpair _)).
rewrite !card_injm ?injm_sdpair1 ?injm_sdpair2 // !cardsT card_prod card_ord.
by rewrite -mulnA Zp_cast.
Qed.
|
Lemma
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
card_pX1p2
| |
Grp_pX1p2:
p^{1+2} \isog Grp (x : y : x ^+ p, y ^+ p, [~ x, y, x], [~ x, y, y]).
Proof.
rewrite [@gtype _]unlock; apply: intro_isoGrp => [|rT H].
apply/existsP; pose x := sdpair1 actp (0, 1)%R; pose y := sdpair2 actp 1%R.
exists (x, y); rewrite /= !xpair_eqE; set z := [~ x, y]; set G := _ <*> _.
have def_z: z = sdpair1 actp (1, 0)%R.
rewrite [z]commgEl -sdpair_act ?inE //=.
rewrite -morphV -?morphM ?inE //=; congr (sdpair1 _ (_, _)) => /=.
by rewrite mulr1 mulKg.
by rewrite mulVg.
have def_xi i: x ^+ i = sdpair1 actp (0, i%:R)%R.
rewrite -morphX ?inE //; congr (sdpair1 _ _).
by apply/eqP; rewrite /eq_op /= !morphX ?inE ?expg1n //=.
have def_yi i: y ^+ i = sdpair2 actp i%:R.
by rewrite -morphX ?inE //.
have def_zi i: z ^+ i = sdpair1 actp (i%:R, 0)%R.
rewrite def_z -morphX ?inE //; congr (sdpair1 _ _).
by apply/eqP; rewrite /eq_op /= !morphX ?inE ?expg1n ?andbT //=.
rewrite def_xi def_yi pchar_Zp ?morph1 //.
rewrite def_z -morphR ?inE // !commgEl -sdpair_act ?inE //= mulr0 addr0.
rewrite mulVg -[_ * _]/(_ , _) /= !invg1 mulg1 !mul1g mulVg morph1 !andbT.
have Gx: x \in G by rewrite -cycle_subG joing_subl.
have Gy: y \in G by rewrite -cycle_subG joing_subr.
rewrite eqEsubset subsetT -im_sdpair mulG_subG /= -/G; apply/andP; split.
apply/subsetP=> u /morphimP[[i j] _ _ def_u].
suffices ->: u = z ^+ i * x ^+ j.
rewrite groupMl; apply/groupX; first exact: Gx.
by apply/groupR; first exact: Gx.
rewrite def_
...
|
Lemma
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
Grp_pX1p2
| |
pX1p2_pgroup: p.-group p^{1+2}.
Proof. by rewrite /pgroup card_pX1p2 pnatX pnat_id. Qed.
|
Lemma
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
pX1p2_pgroup
| |
pX1p2_extraspecial: extraspecial p^{1+2}.
Proof.
apply: (p3group_extraspecial pX1p2_pgroup); last first.
by rewrite card_pX1p2 pfactorK.
case/existsP: (isoGrp_hom Grp_pX1p2) card_pX1p2 => [[x y]] /=.
case/eqP=> <- xp yp _ _ oXY.
apply: contraL (dvdn_cardMg <[x]> <[y]>) => cXY_XY.
rewrite -cent_joinEl ?(sub_abelian_cent2 cXY_XY) ?joing_subl ?joing_subr //.
rewrite oXY -!orderE pfactor_dvdn ?muln_gt0 ?order_gt0 // -leqNgt.
rewrite -(pfactorK 2 p_pr) dvdn_leq_log ?expn_gt0 ?p_gt0 //.
by rewrite dvdn_mul ?order_dvdn ?xp ?yp.
Qed.
|
Lemma
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
pX1p2_extraspecial
| |
exponent_pX1p2: odd p -> exponent p^{1+2} %| p.
Proof.
move=> p_odd; have pG := pX1p2_pgroup.
have ->: p^{1+2} = 'Ohm_1(p^{1+2}).
apply/eqP; rewrite eqEsubset Ohm_sub andbT (OhmE 1 pG).
case/existsP: (isoGrp_hom Grp_pX1p2) => [[x y]] /=.
case/eqP=> <- xp yp _ _; rewrite joing_idl joing_idr genS //.
by rewrite subsetI subset_gen subUset !sub1set !inE xp yp!eqxx.
rewrite exponent_Ohm1_class2 ?card_pX1p2 ?oddX // nil_class2.
by have [[_ ->] _ ] := pX1p2_extraspecial.
Qed.
|
Lemma
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
exponent_pX1p2
| |
isog_pX1p2(gT : finGroupType) (G : {group gT}) :
extraspecial G -> exponent G %| p -> #|G| = (p ^ 3)%N -> G \isog p^{1+2}.
Proof.
move=> esG expGp oG; apply/(isoGrpP _ Grp_pX1p2).
rewrite card_pX1p2; split=> //.
have pG: p.-group G by rewrite /pgroup oG pnatX pnat_id.
have oZ := card_center_extraspecial pG esG.
have [x Gx notZx]: exists2 x, x \in G & x \notin 'Z(G).
apply/subsetPn; rewrite proper_subn // properEcard center_sub oZ oG.
by rewrite (ltn_exp2l 1 3).
have ox: #[x] = p.
by apply: nt_prime_order; rewrite ?(exponentP expGp) ?(group1_contra notZx).
have [y Gy not_cxy]: exists2 y, y \in G & y \notin 'C[x].
by apply/subsetPn; rewrite sub_cent1; rewrite inE Gx in notZx.
apply/existsP; exists (x, y) => /=; set z := [~ x, y].
have [[defPhiG defG'] _] := esG.
have Zz: z \in 'Z(G) by rewrite -defG' mem_commg.
have [Gz cGz] := setIP Zz; rewrite !xpair_eqE !(exponentP expGp) //.
have [_ nZG] := andP (center_normal G).
rewrite /commg /conjg !(centP cGz) // !mulKg mulVg !eqxx !andbT.
have sXY_G: <[x]> <*> <[y]> \subset G by rewrite join_subG !cycle_subG Gx.
have defZ: <[z]> = 'Z(G).
apply/eqP; rewrite eqEcard cycle_subG Zz oZ /= -orderE.
rewrite (nt_prime_order p_pr) ?(exponentP expGp) //.
by rewrite (sameP commgP cent1P) cent1C.
have sZ_XY: 'Z(G) \subset <[x]> <*> <[y]>.
by rewrite -defZ cycle_subG groupR // mem_gen // inE cycle_id ?orbT.
rewrite eqEcard sXY_G /= oG -(Lagrange sZ_XY) oZ leq_pmul2l //.
rewrite -card_quotient ?(subset_trans sXY_G) //.
rewrite quotientY ?qu
...
|
Lemma
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
isog_pX1p2
| |
pX1p2id: p^{1+2*1} \isog p^{1+2}.
Proof. exact: ncprod1. Qed.
|
Lemma
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
pX1p2id
| |
pX1p2Sn : xcprod_spec p^{1+2} p^{1+2*n} p^{1+2*n.+1}%type.
Proof. exact: ncprodS. Qed.
|
Lemma
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
pX1p2S
| |
card_pX1p2nn : prime p -> #|p^{1+2*n}| = (p ^ n.*2.+1)%N.
Proof.
move=> p_pr; have pG := pX1p2_pgroup p_pr.
have oG := card_pX1p2 p_pr; have esG := pX1p2_extraspecial p_pr.
have oZ := card_center_extraspecial pG esG.
elim: n => [|n IHn]; first by rewrite (card_isog (ncprod0 _)) oZ.
case: pX1p2S => gz isoZ; rewrite -im_cpair cardMg_divn setI_im_cpair.
rewrite -injm_center ?{1}card_injm ?injm_cpairg1 ?injm_cpair1g ?center_sub //.
by rewrite oG oZ IHn -expnD mulKn ?prime_gt0.
Qed.
|
Lemma
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
card_pX1p2n
| |
pX1p2n_pgroupn : prime p -> p.-group p^{1+2*n}.
Proof. by move=> p_pr; rewrite /pgroup card_pX1p2n // pnatX pnat_id. Qed.
|
Lemma
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
pX1p2n_pgroup
| |
exponent_pX1p2nn : prime p -> odd p -> exponent p^{1+2*n} = p.
Proof.
move=> p_pr odd_p; apply: prime_nt_dvdP => //.
rewrite -dvdn1 -trivg_exponent -cardG_gt1 card_pX1p2n //.
by rewrite (ltn_exp2l 0) // prime_gt1.
elim: n => [|n IHn].
by rewrite (dvdn_trans (exponent_dvdn _)) ?card_pX1p2n.
case: pX1p2S => gz isoZ; rewrite -im_cpair /=.
apply/exponentP=> xy; case/imset2P=> x y C1x C2y ->{xy}.
rewrite expgMn; last by red; rewrite -(centsP (im_cpair_cent isoZ)).
rewrite (exponentP _ y C2y) ?exponent_injm ?injm_cpair1g // mulg1.
by rewrite (exponentP _ x C1x) ?exponent_injm ?injm_cpairg1 // exponent_pX1p2.
Qed.
|
Lemma
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
exponent_pX1p2n
| |
pX1p2n_extraspecialn : prime p -> n > 0 -> extraspecial p^{1+2*n}.
Proof.
move=> p_pr; elim: n => [//|n IHn _].
have esG := pX1p2_extraspecial p_pr.
have [n0 | n_gt0] := posnP n.
by apply: isog_extraspecial esG; rewrite isog_sym n0 pX1p2id.
case: pX1p2S (pX1p2n_pgroup n.+1 p_pr) => gz isoZ pGn.
apply: (cprod_extraspecial pGn (im_cpair_cprod isoZ) (setI_im_cpair isoZ)).
by apply: injm_extraspecial esG; rewrite ?injm_cpairg1.
by apply: injm_extraspecial (IHn n_gt0); rewrite ?injm_cpair1g.
Qed.
|
Lemma
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
pX1p2n_extraspecial
| |
Ohm1_extraspecial_odd(gT : finGroupType) (G : {group gT}) :
p.-group G -> extraspecial G -> odd #|G| ->
let Y := 'Ohm_1(G) in
[/\ exponent Y = p, #|G : Y| %| p
& Y != G ->
exists E : {group gT},
[/\ #|G : Y| = p, #|E| = p \/ extraspecial E,
exists2 X : {group gT}, #|X| = p & X \x E = Y
& exists M : {group gT},
[/\ M \isog 'Mod_(p ^ 3), M \* E = G & M :&: E = 'Z(M)]]].
Proof.
move=> pG esG oddG Y; have [spG _] := esG.
have [defPhiG defG'] := spG; set Z := 'Z(G) in defPhiG defG'.
have{spG} expG: exponent G %| p ^ 2 by apply: exponent_special.
have p_pr := extraspecial_prime pG esG.
have p_gt1 := prime_gt1 p_pr; have p_gt0 := ltnW p_gt1.
have oZ: #|Z| = p := card_center_extraspecial pG esG.
have nsZG: Z <| G := center_normal G; have [sZG nZG] := andP nsZG.
have nsYG: Y <| G := Ohm_normal 1 G; have [sYG nYG] := andP nsYG.
have ntZ: Z != 1 by rewrite -cardG_gt1 oZ.
have sZY: Z \subset Y.
by apply: contraR ntZ => ?; rewrite -(setIidPl sZG) TI_Ohm1 ?prime_TIg ?oZ.
have ntY: Y != 1 by apply: subG1_contra ntZ.
have p_odd: odd p by rewrite -oZ (oddSg sZG).
have expY: exponent Y %| p by rewrite exponent_Ohm1_class2 // nil_class2 defG'.
rewrite (prime_nt_dvdP p_pr _ expY) -?dvdn1 -?trivg_exponent //.
have [-> | neYG] := eqVneq Y G; first by rewrite indexgg dvd1n; split.
have sG1Z: 'Mho^1(G) \subset Z by rewrite -defPhiG (Phi_joing pG) joing_subr.
have Z_Gp: {in G, forall x, x ^+ p \in Z}.
by move=> x Gx; rewrite /= (subsetP sG1Z) ?(Mho_p_elt 1)
...
|
Lemma
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
Ohm1_extraspecial_odd
| |
isog_pX1p2nn (gT : finGroupType) (G : {group gT}) :
prime p -> extraspecial G -> #|G| = (p ^ n.*2.+1)%N -> exponent G %| p ->
G \isog p^{1+2*n}.
Proof.
move=> p_pr esG oG expG; have p_gt1 := prime_gt1 p_pr.
have not_le_p3_p: ~~ (p ^ 3 <= p) by rewrite (leq_exp2l 3 1).
have pG: p.-group G by rewrite /pgroup oG pnatX pnat_id.
have oZ := card_center_extraspecial pG esG.
have{pG esG} [Es p3Es defG] := extraspecial_structure pG esG.
set Z := 'Z(G) in oZ defG p3Es.
elim: Es {+}G => [|E Es IHs] S in n oG expG p3Es defG *.
rewrite big_nil cprod1g in defG; rewrite -defG.
have ->: n = 0.
apply: double_inj; apply/eqP.
by rewrite -eqSS -(eqn_exp2l _ _ p_gt1) -oG -defG oZ.
by rewrite isog_cyclic_card prime_cyclic ?oZ ?card_pX1p2n //=.
rewrite big_cons -cprodA in defG; rewrite /= -andbA in p3Es.
have [[_ T _ defT] defET cTE] := cprodP defG; rewrite defT in defET cTE defG.
move: p3Es => /and3P[/eqP oE /eqP defZE /IHs{}IHs].
have not_cEE: ~~ abelian E.
by apply: contra not_le_p3_p => cEE; rewrite -oE -oZ -defZE (center_idP _).
have sES: E \subset S by rewrite -defET mulG_subl.
have sTS: T \subset S by rewrite -defET mulG_subr.
have expE: exponent E %| p by apply: dvdn_trans (exponentS sES) expG.
have expT: exponent T %| p by apply: dvdn_trans (exponentS sTS) expG.
have{expE not_cEE} isoE: E \isog p^{1+2}.
apply: isog_pX1p2 => //.
by apply: card_p3group_extraspecial p_pr oE _; rewrite defZE.
have sZT: 'Z(E) \subset T.
by case/cprodP: defT => [[U _ -> _] <- _]; rewrite defZE m
...
|
Lemma
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
isog_pX1p2n
| |
isog_2X1p2: 2^{1+2} \isog 'D_8.
Proof.
have pr2: prime 2 by []; have oG := card_pX1p2 pr2; rewrite -[8]oG.
case/existsP: (isoGrp_hom (Grp_pX1p2 pr2)) => [[x y]] /=.
rewrite -/2^{1+2}; case/eqP=> defG x2 y2 _ _.
have not_oG_2: ~~ (#|2^{1+2}| %| 2) by rewrite oG.
have ox: #[x] = 2.
apply: nt_prime_order => //; apply: contra not_oG_2 => x1.
by rewrite -defG (eqP x1) cycle1 joing1G order_dvdn y2.
have oy: #[y] = 2.
apply: nt_prime_order => //; apply: contra not_oG_2 => y1.
by rewrite -defG (eqP y1) cycle1 joingG1 order_dvdn x2.
rewrite -defG joing_idl joing_idr involutions_gen_dihedral //.
apply: contra not_oG_2 => eq_xy; rewrite -defG (eqP eq_xy) (joing_idPl _) //.
by rewrite -orderE oy.
Qed.
|
Lemma
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
isog_2X1p2
| |
Q8_extraspecial: extraspecial 'Q_8.
Proof.
have gt32: 3 > 2 by []; have isoQ: 'Q_8 \isog 'Q_(2 ^ 3) by apply: isog_refl.
have [[x y] genQ _] := generators_quaternion gt32 isoQ.
have [_ [defQ' defPhiQ _ _]] := quaternion_structure gt32 genQ isoQ.
case=> defZ oZ _ _ _ _ _; split; last by rewrite oZ.
by split; rewrite ?defPhiQ defZ.
Qed.
|
Lemma
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
Q8_extraspecial
| |
DnQ_Pn : xcprod_spec 'D^n 'Q_8 ('D^n*Q)%type.
Proof.
have pQ: 2.-group 'Q_(2 ^ 3) by rewrite /pgroup card_quaternion.
have{pQ} oZQ := card_center_extraspecial pQ Q8_extraspecial.
suffices oZDn: #|'Z('D^n)| = 2.
by apply: xcprodP; rewrite isog_cyclic_card ?prime_cyclic ?oZQ ?oZDn.
have [-> | n_gt0] := posnP n; first by rewrite center_ncprod0 card_pX1p2n.
have pr2: prime 2 by []; have pDn := pX1p2n_pgroup n pr2.
exact: card_center_extraspecial (pX1p2n_extraspecial pr2 n_gt0).
Qed.
|
Lemma
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
DnQ_P
| |
card_DnQn : #|'D^n*Q| = (2 ^ n.+1.*2.+1)%N.
Proof.
have oQ: #|'Q_(2 ^ 3)| = 8 by rewrite card_quaternion.
have pQ: 2.-group 'Q_8 by rewrite /pgroup oQ.
case: DnQ_P => gz isoZ.
rewrite -im_cpair cardMg_divn setI_im_cpair cpair_center_id.
rewrite -injm_center//; last exact: injm_cpair1g.
rewrite (card_injm (injm_cpairg1 _))//= (card_injm (injm_cpair1g _))//.
rewrite (card_injm (injm_cpair1g _))//; last exact: center_sub.
rewrite oQ card_pX1p2n // (card_center_extraspecial pQ Q8_extraspecial).
by rewrite -muln_divA // mulnC -(expnD 2 2).
Qed.
|
Lemma
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
card_DnQ
| |
DnQ_pgroupn : 2.-group 'D^n*Q.
Proof. by rewrite /pgroup card_DnQ pnatX. Qed.
|
Lemma
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
DnQ_pgroup
| |
DnQ_extraspecialn : extraspecial 'D^n*Q.
Proof.
case: DnQ_P (DnQ_pgroup n) => gz isoZ pDnQ.
have [injDn injQ] := (injm_cpairg1 isoZ, injm_cpair1g isoZ).
have [n0 | n_gt0] := posnP n.
rewrite -im_cpair mulSGid; first exact: injm_extraspecial Q8_extraspecial.
apply/setIidPl; rewrite setI_im_cpair -injm_center //=.
by congr (_ @* _); rewrite n0 center_ncprod0.
apply: (cprod_extraspecial pDnQ (im_cpair_cprod isoZ) (setI_im_cpair _)).
exact: injm_extraspecial (pX1p2n_extraspecial _ _).
exact: injm_extraspecial Q8_extraspecial.
Qed.
|
Lemma
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
DnQ_extraspecial
| |
card_isog8_extraspecial(gT : finGroupType) (G : {group gT}) :
#|G| = 8 -> extraspecial G -> (G \isog 'D_8) || (G \isog 'Q_8).
Proof.
move=> oG esG; have pG: 2.-group G by rewrite /pgroup oG.
apply/norP=> [[notG_D8 notG_Q8]].
have not_extG: extremal_class G = NotExtremal.
by rewrite /extremal_class oG andFb (negPf notG_D8) (negPf notG_Q8).
have [x Gx ox] := exponent_witness (pgroup_nil pG).
pose X := <[x]>; have cycX: cyclic X := cycle_cyclic x.
have sXG: X \subset G by rewrite cycle_subG.
have iXG: #|G : X| = 2.
by rewrite -divgS // oG -orderE -ox exponent_2extraspecial.
have not_cGG := extraspecial_nonabelian esG.
have:= maximal_cycle_extremal pG not_cGG cycX sXG iXG.
by rewrite /extremal2 not_extG.
Qed.
|
Lemma
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
card_isog8_extraspecial
| |
isog_2extraspecial(gT : finGroupType) (G : {group gT}) n :
#|G| = (2 ^ n.*2.+1)%N -> extraspecial G -> G \isog 'D^n \/ G \isog 'D^n.-1*Q.
Proof.
elim: n G => [|n IHn] G oG esG.
case/negP: (extraspecial_nonabelian esG).
by rewrite cyclic_abelian ?prime_cyclic ?oG.
have pG: 2.-group G by rewrite /pgroup oG pnatX.
have oZ:= card_center_extraspecial pG esG.
have: 'Z(G) \subset 'Ohm_1(G).
apply/subsetP=> z Zz; rewrite (OhmE _ pG) mem_gen //.
by rewrite !inE -order_dvdn -oZ order_dvdG ?(subsetP (center_sub G)).
rewrite subEproper; case/predU1P=> [defG1 | ltZG1].
have [n' n'_gt2 isoG]: exists2 n', n' > 2 & G \isog 'Q_(2 ^ n').
apply/quaternion_classP; apply/eqP.
have not_cycG: ~~ cyclic G.
by apply: contra (extraspecial_nonabelian esG); apply: cyclic_abelian.
move: oZ; rewrite defG1; move/prime_Ohm1P; rewrite (negPf not_cycG) /=.
by apply=> //; apply: contra not_cycG; move/eqP->; apply: cyclic1.
have [n0 n'3]: n = 0 /\ n' = 3.
have [[x y] genG _] := generators_quaternion n'_gt2 isoG.
have n'3: n' = 3.
have [_ [_ _ oG' _] _ _ _] := quaternion_structure n'_gt2 genG isoG.
apply/eqP; rewrite -(subnKC (ltnW n'_gt2)) subn2 !eqSS -(@eqn_exp2l 2) //.
by rewrite -oG' -oZ; case: esG => [[_ ->]].
by move/eqP: oG; have [-> _ _ _] := genG; rewrite n'3 eqn_exp2l //; case n.
right; rewrite (isog_trans isoG) // n'3 n0 /=.
case: DnQ_P => z isoZ; rewrite -im_cpair mulSGid ?sub_isog ?injm_cpair1g //.
apply/setIidPl; rewrite setI_im_cpair -injm_cent
...
|
Lemma
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
isog_2extraspecial
| |
rank_Dnn : 'r_2('D^n) = n.+1.
Proof.
elim: n => [|n IHn]; first by rewrite p_rank_abelem ?prime_abelem ?card_pX1p2n.
have oDDn: #|'D^n.+1| = (2 ^ n.+1.*2.+1)%N by apply: card_pX1p2n.
have esDDn: extraspecial 'D^n.+1 by apply: pX1p2n_extraspecial.
do [case: pX1p2S => gz isoZ; set DDn := [set: _]] in oDDn esDDn *.
have pDDn: 2.-group DDn by rewrite /pgroup oDDn pnatX.
apply/eqP; rewrite eqn_leq; apply/andP; split.
have [E EprE]:= p_rank_witness 2 [group of DDn].
have [sEDDn abelE <-] := pnElemP EprE; have [pE cEE _]:= and3P abelE.
rewrite -(@leq_exp2l 2) // -p_part part_pnat_id // -leq_sqr -expnM -mulnn.
rewrite muln2 doubleS expnS -oDDn -(@leq_pmul2r #|'C_DDn(E)|) ?cardG_gt0 //.
rewrite {1}(card_subcent_extraspecial pDDn) // mulnCA -mulnA Lagrange //=.
rewrite mulnAC mulnA leq_pmul2r ?cardG_gt0 // setTI.
have ->: (2 * #|'C(E)| = #|'Z(DDn)| * #|'C(E)|)%N.
by rewrite (card_center_extraspecial pDDn).
by rewrite leq_mul ?subset_leq_card ?subsetIl.
have [inj1 injn] := (injm_cpairg1 isoZ, injm_cpair1g isoZ).
pose D := cpairg1 isoZ @* 2^{1+2}; pose Dn := cpair1g isoZ @* 'D^n.
have [E EprE] := p_rank_witness 2 [group of Dn].
rewrite injm_p_rank //= IHn in EprE; have [sEDn abelE dimE]:= pnElemP EprE.
have [x [Dx ox] notDnx]: exists x, [/\ x \in D, #[x] = 2 & x \notin Dn].
have isoD: D \isog 'D_(2 ^ 3).
by rewrite isog_sym -(isog_transl _ isog_2X1p2) sub_isog.
have [//| [x y] genD [oy _]] := generators_2dihedral _ isoD.
have [_ _ _ X'y] := genD; case/setDP: X'y; r
...
|
Lemma
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
rank_Dn
| |
rank_DnQn : 'r_2('D^n*Q) = n.+1.
Proof.
have pDnQ: 2.-group 'D^n*Q := DnQ_pgroup n.
have esDnQ: extraspecial 'D^n*Q := DnQ_extraspecial n.
do [case: DnQ_P => gz isoZ; set DnQ := setT] in pDnQ esDnQ *.
suffices [E]: exists2 E, E \in 'E*_2(DnQ) & logn 2 #|E| = n.+1.
by rewrite (pmaxElem_extraspecial pDnQ esDnQ); case/pnElemP=> _ _ <-.
have oZ: #|'Z(DnQ)| = 2 by apply: card_center_extraspecial.
pose Dn := cpairg1 isoZ @* 'D^n; pose Q := cpair1g isoZ @* 'Q_8.
have [injDn injQ] := (injm_cpairg1 isoZ, injm_cpair1g isoZ).
have [E EprE]:= p_rank_witness 2 [group of Dn].
have [sEDn abelE dimE] := pnElemP EprE; have [pE cEE eE]:= and3P abelE.
rewrite injm_p_rank // rank_Dn in dimE; exists E => //.
have sZE: 'Z(DnQ) \subset E.
have maxE := subsetP (p_rankElem_max _ _) E EprE.
have abelZ: 2.-abelem 'Z(DnQ) by rewrite prime_abelem ?oZ.
rewrite -(Ohm1_id abelZ) (OhmE _ (abelem_pgroup abelZ)) gen_subG.
rewrite -(pmaxElem_LdivP _ maxE) // setSI //=.
by rewrite -cpairg1_center injm_center // setIS ?centS.
have scE: 'C_Dn(E) = E.
apply/eqP; rewrite eq_sym eqEcard subsetI sEDn -abelianE cEE /=.
have [n0 | n_gt0] := posnP n.
rewrite subset_leq_card // subIset // (subset_trans _ sZE) //.
by rewrite -cpairg1_center morphimS // n0 center_ncprod0.
have pDn: 2.-group Dn by rewrite morphim_pgroup ?pX1p2n_pgroup.
have esDn: extraspecial Dn.
exact: injm_extraspecial (pX1p2n_extraspecial _ _).
rewrite dvdn_leq ?cardG_gt0 // (card_subcent_extraspecial pDn) //=.
rewrite -injm_ce
...
|
Lemma
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
rank_DnQ
| |
not_isog_Dn_DnQn : ~~ ('D^n \isog 'D^n.-1*Q).
Proof.
case: n => [|n] /=; first by rewrite isogEcard card_pX1p2n // card_DnQ andbF.
apply: contraL (leqnn n.+1) => isoDn1DnQ.
by rewrite -ltnNge -rank_Dn (isog_p_rank isoDn1DnQ) rank_DnQ.
Qed.
|
Lemma
|
solvable
|
[
"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 finalg zmodp cyclic pgroup center gseries",
"From mathcomp Require Import nilpotent sylow abelian finmodule matrix maximal",
"From mathcomp Require Import extremal"
] |
solvable/extraspecial.v
|
not_isog_Dn_DnQ
| |
aut_of:=
odflt 1 [pick s in Aut B | p > 1 & (#[s] %| p) && (s b == b ^+ e)].
|
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
|
aut_of
| |
aut_dvdn: #[aut_of] %| #[a].
Proof.
rewrite order_Zp1 /aut_of; case: pickP => [s | _]; last by rewrite order1.
by case/and4P=> _ p_gt1 p_s _; rewrite Zp_cast.
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
|
aut_dvdn
| |
act_morphism:= eltm_morphism aut_dvdn.
|
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
|
act_morphism
| |
base_act:= ([Aut B] \o act_morphism)%gact.
|
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
|
base_act
| |
act_dom: <[a]> \subset act_dom base_act.
Proof.
rewrite cycle_subG 2!inE cycle_id /= eltm_id /aut_of.
by case: pickP => [op /andP[] | _] //=; rewrite group1.
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
|
act_dom
| |
gact:= (base_act \ act_dom)%gact.
|
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
|
gact
| |
gtype_unlockable:= Unlockable gtype.unlock.
|
Canonical
|
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
|
gtype_unlockable
| |
card: #|[set: gtype]| = (p * q)%N.
Proof.
rewrite [gtype.body]unlock -(sdprod_card (sdprod_sdpair _)).
rewrite !card_injm ?injm_sdpair1 ?injm_sdpair2 //.
by rewrite mulnC -!orderE !order_Zp1 !Zp_cast.
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
| |
Grp: (exists s, [/\ s \in Aut B, #[s] %| p & s b = b ^+ e]) ->
[set: gtype] \isog Grp (x : y : x ^+ q, y ^+ p, x ^ y = x ^+ e).
Proof.
rewrite [gtype.body]unlock => [[s [AutBs dvd_s_p sb]]].
have memB: _ \in B by move=> c; rewrite -Zp_cycle inE.
have Aa: a \in <[a]> by rewrite !cycle_id.
have [oa ob]: #[a] = p /\ #[b] = q by rewrite !order_Zp1 !Zp_cast.
have def_s: aut_of = s.
rewrite /aut_of; case: pickP => /= [t | ]; last first.
by move/(_ s); case/and4P; rewrite sb.
case/and4P=> AutBt _ _ tb; apply: (eq_Aut AutBt) => // b_i.
case/cycleP=> i ->; rewrite -(autmE AutBt) -(autmE AutBs) !morphX //=.
by rewrite !autmE // sb (eqP tb).
apply: intro_isoGrp => [|gT G].
apply/existsP; exists (sdpair1 _ b, sdpair2 _ a); rewrite /= !xpair_eqE.
apply/andP; split.
by rewrite -!morphim_cycle ?norm_joinEr ?im_sdpair ?im_sdpair_norm ?eqxx //=.
rewrite -!order_dvdn !order_injm ?injm_sdpair1 ?injm_sdpair2 // oa ob !dvdnn.
by rewrite -sdpair_act // [act _ _ _]apermE /= eltm_id -morphX // -sb -def_s.
case/existsP=> -[x y] /= /eqP[defG xq1 yp1 xy].
have fxP: #[x] %| #[b] by rewrite order_dvdn ob xq1.
have fyP: #[y] %| #[a] by rewrite order_dvdn oa yp1.
have fP: {in <[b]> & <[a]>, morph_act gact 'J (eltm fxP) (eltm fyP)}.
move=> bj ai; case/cycleP=> j ->{bj}; case/cycleP=> i ->{ai}.
rewrite /= !eltmE def_s gactX ?groupX // conjXg morphX //=; congr (_ ^+ j).
rewrite /autact /= apermE; elim: i {j} => /= [|i IHi].
by rewrite perm1 eltm_id conjg1.
rewrite !expgS per
...
|
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_gtype:= gtype q p (q %/ p).+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_gtype
| |
dihedral_gtype:= gtype q 2 q.-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
|
dihedral_gtype
| |
semidihedral_gtype:= gtype q 2 (q %/ p).-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
|
semidihedral_gtype
| |
quaternion_kernel:=
<<[set u | u ^+ 2 == 1] :\: [set u ^+ 2 | u in [set: gtype q 4 q.-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
|
quaternion_kernel
| |
quaternion_unlock:= Unlockable quaternion_gtype.unlock.
|
Canonical
|
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_unlock
| |
cyclic_pgroup_Aut_structuregT p (G : {group gT}) :
p.-group G -> cyclic G -> G :!=: 1 ->
let q := #|G| in let n := (logn p q).-1 in
let A := Aut G in let P := 'O_p(A) in let F := 'O_p^'(A) in
exists m : {perm gT} -> 'Z_q,
[/\ [/\ {in A & G, forall a x, x ^+ m a = a x},
m 1 = 1%R /\ {in A &, {morph m : a b / a * b >-> (a * b)%R}},
{in A &, injective m} /\ image m A =i GRing.unit,
forall k, {in A, {morph m : a / a ^+ k >-> (a ^+ k)%R}}
& {in A, {morph m : a / a^-1 >-> (a^-1)%R}}],
[/\ abelian A, cyclic F, #|F| = p.-1
& [faithful F, on 'Ohm_1(G) | [Aut G]]]
& if n == 0 then A = F else
exists t, [/\ t \in A, #[t] = 2, m t = (- 1)%R
& if odd p then
[/\ cyclic A /\ cyclic P,
exists s, [/\ s \in A, #[s] = (p ^ n)%N, m s = p.+1%:R & P = <[s]>]
& exists s0, [/\ s0 \in A, #[s0] = p, m s0 = (p ^ n).+1%:R
& 'Ohm_1(P) = <[s0]>]]
else if n == 1%N then A = <[t]>
else exists s,
[/\ s \in A, #[s] = (2 ^ n.-1)%N, m s = 5%:R, <[s]> \x <[t]> = A
& exists s0, [/\ s0 \in A, #[s0] = 2, m s0 = (2 ^ n).+1%:R,
m (s0 * t) = (2 ^ n).-1%:R & 'Ohm_1(<[s]>) = <[s0]>]]]].
Proof.
move=> pG cycG ntG q n0 A P F; have [p_pr p_dvd_G [n oG]] := pgroup_pdiv pG ntG.
have [x0 defG] := cyclicP cycG; have Gx0: x0 \in G by rewrite defG cycle_id.
rewrite {1}/q oG pfactorK //= in n0 *; rewrite {}/n0.
have [p_gt1 min_p] := primeP p_pr; have p_gt0 := ltnW p_gt1.
have q_gt1: q >
...
|
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_pgroup_Aut_structure
| |
extremal_generatorsgT (A : {set gT}) p n xy :=
let: (x, y) := xy in
[/\ #|A| = (p ^ n)%N, x \in A, #[x] = (p ^ n.-1)%N & y \in A :\: <[x]>].
|
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_generators
| |
extremal_generators_factsgT (G : {group gT}) p n x y :
prime p -> extremal_generators G p n (x, y) ->
[/\ p.-group G, maximal <[x]> G, <[x]> <| G,
<[x]> * <[y]> = G & <[y]> \subset 'N(<[x]>)].
Proof.
move=> p_pr [oG Gx ox] /setDP[Gy notXy].
have pG: p.-group G by rewrite /pgroup oG pnatX pnat_id.
have maxX: maximal <[x]> G.
rewrite p_index_maximal -?divgS ?cycle_subG // -orderE oG ox.
case: (n) oG => [|n' _]; last by rewrite -expnB ?subSnn ?leqnSn ?prime_gt0.
move/eqP; rewrite -trivg_card1; case/trivgPn.
by exists y; rewrite // (group1_contra notXy).
have nsXG := p_maximal_normal pG maxX; split=> //.
by apply: mulg_normal_maximal; rewrite ?cycle_subG.
by rewrite cycle_subG (subsetP (normal_norm 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
|
extremal_generators_facts
| |
card_modular_group: #|'Mod_(p ^ n)| = (p ^ n)%N.
Proof. by rewrite Extremal.card def_p ?def_q // -expnS def_n. 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_modular_group
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.