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 fin... | 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... | 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 fin... | 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_... | 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 fin... | 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 fin... | 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 morp... | 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 fin... | 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 fin... | 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 fin... | 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 fin... | 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 fin... | 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 fin... | 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 fin... | 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 fin... | 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 fin... | 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.
Qe... | 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 fin... | 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 fin... | 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 fin... | 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 fin... | 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 fin... | 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 fin... | 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) ?nor... | 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 fin... | 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 fin... | 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 fin... | 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 fin... | 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 fin... | 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 fin... | 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 fin... | 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_... | 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 fin... | 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 fin... | 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 ... | 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 fin... | 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_pmul... | 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 fin... | 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]... | 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 fin... | 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 fin... | 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 fin... | 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 fin... | 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: cycl... | 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 fin... | 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 fin... | 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 fin... | 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 fin... | 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);... | 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 fin... | 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: ... | 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 fin... | 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_ca... | 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 fin... | 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;... | 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 fin... | 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: co... | 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 fin... | 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 fin... | 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 fin... | 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 ... | 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 fin... | 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/... | 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 fin... | 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 ... | 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 fin... | 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 fin... | 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"... | 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"... | 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"... | 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"... | 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"... | 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"... | 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"... | 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"... | 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"... | 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 =... | 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"... | 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"... | 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_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"... | 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 s... | 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"... | 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 ... | 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"... | 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"... | 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"... | 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_... | 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"... | 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"... | 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... | 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"... | 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_extraspe... | 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"... | 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 ... | 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"... | 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 pna... | 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"... | 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: c... | 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"... | 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 ?de... | 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"... | 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 b... | 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"... | 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 ... | 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"... | 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"... | 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 //=.
... | 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"... | 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 n... | 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"... | 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.
h... | 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"... | 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 p... | 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"... | 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/p... | 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"... | 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"... | 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 quotie... | 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 quotie... | 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 quotie... | 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 quotie... | 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 quotie... | 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 quotie... | 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 quotie... | 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 quotie... | 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... | 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 quotie... | 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 quotie... | 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 quotie... | 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 quotie... | 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 quotie... | 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 quotie... | 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... | 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 quotie... | 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 quotie... | 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:... | 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 quotie... | 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 quotie... | solvable/extremal.v | card_modular_group |
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