statement stringlengths 1 4.33k | proof stringlengths 0 37.9k | type stringclasses 25
values | symbolic_name stringlengths 1 67 | library stringclasses 10
values | filename stringclasses 112
values | imports listlengths 2 138 | deps listlengths 0 64 | docstring stringclasses 798
values | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
morph_order : #[f x] %| #[x]. | Proof. by rewrite order_dvdn -morphX // expg_order morph1. Qed. | Lemma | morph_order | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"expg_order",
"morph1",
"morphX",
"order_dvdn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morph_generator A : generator A x -> generator (f @* A) (f x). | Proof. by move/(A =P _)->; rewrite /generator morphim_cycle. Qed. | Lemma | morph_generator | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"generator",
"morphim_cycle"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cyclicS G H : H \subset G -> cyclic G -> cyclic H. | Proof.
move=> sHG /cyclicP[x defG]; apply/cyclicP.
exists (x ^+ (#[x] %/ #|H|)); apply/congr_group/set1P.
by rewrite -cycle_sub_group /order -defG ?cardSg // inE sHG eqxx.
Qed. | Lemma | cyclicS | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"cardSg",
"congr_group",
"cycle_sub_group",
"cyclic",
"cyclicP",
"defG",
"eqxx",
"inE",
"order",
"sHG",
"set1P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cyclicJ G 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 | cyclicJ | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"conjsgK",
"cycleJ",
"cyclic",
"cyclicP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_subG_cyclic G 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 | eq_subG_cyclic | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"cardSg",
"cycle_sub_group",
"cyclic",
"cyclicP",
"inE",
"set1P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cardSg_cyclic G 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... | Lemma | cardSg_cyclic | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"cardSg",
"cycleX",
"cyclic",
"cyclicP",
"cyclicS",
"dvdnP",
"dvdn_mulr",
"eq_subG_cyclic",
"last",
"mulKn",
"orderE",
"orderXdiv",
"order_gt0",
"sHG",
"sKG",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_cyclic_char G H : cyclic G -> (H \char G) = (H \subset G). | Proof.
case/cyclicP=> x ->; apply/idP/idP => [/andP[] //|].
exact: cycle_subgroup_char.
Qed. | Lemma | sub_cyclic_char | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"char",
"cycle_subgroup_char",
"cyclic",
"cyclicP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_cyclic rT 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 | morphim_cyclic | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"cyclic",
"cyclicP",
"cyclicS",
"gen_subG",
"morphimIdom",
"morphim_cycle",
"morphism",
"sHG",
"sub1set",
"subsetIl",
"subsetIr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quotient_cycle x H : x \in 'N(H) -> <[x]> / H = <[coset H x]>. | Proof. exact: morphim_cycle. Qed. | Lemma | quotient_cycle | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"coset",
"morphim_cycle"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quotient_cyclic G H : cyclic G -> cyclic (G / H). | Proof. exact: morphim_cyclic. Qed. | Lemma | quotient_cyclic | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"cyclic",
"morphim_cyclic"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quotient_generator x G H :
x \in 'N(H) -> generator G x -> generator (G / H) (coset H x). | Proof. by move=> Nx; apply: morph_generator. Qed. | Lemma | quotient_generator | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"coset",
"generator",
"morph_generator"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prime_cyclic G : 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 | prime_cyclic | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"cycle_subG",
"cyclic",
"cyclicP",
"eqEcard",
"eq_sym",
"eqnP",
"leqnn",
"ltnNge",
"order_dvdG",
"order_eq1",
"prime",
"primeP",
"trivgPn",
"trivg_card_le1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_prime_cyclic G 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 | dvdn_prime_cyclic | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"cyclic",
"cyclic1",
"eqsVneq",
"pG",
"p_pr",
"prime",
"prime_cyclic",
"prime_nt_dvdP",
"trivg_card1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cyclic_small G : #|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 | cyclic_small | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"cyclic",
"cyclic1",
"leq_eqVlt",
"ltnS",
"pred2P",
"predU1P",
"prime_cyclic",
"trivg_card_le1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
injm_cyclic G 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 | injm_cyclic | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"cyclic",
"injf",
"last",
"morphim_cyclic",
"morphim_invm",
"morphism",
"sHG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isog_cyclic G M : G \isog M -> cyclic G = cyclic M. | Proof. by case/isogP=> f injf <-; rewrite injm_cyclic. Qed. | Lemma | isog_cyclic | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"cyclic",
"injf",
"injm_cyclic",
"isog",
"isogP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isog_cyclic_card G 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 | isog_cyclic_card | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"Zp_isog",
"apply",
"card_isog",
"cyclic",
"cyclicP",
"isog",
"isog_cyclic",
"isog_symr",
"isog_trans",
"order"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
injm_generator G 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 | injm_generator | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"generator",
"injf",
"invmE",
"last",
"mem_morphim",
"morph_generator",
"morphim_invm",
"morphism",
"sHG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
metacyclic A | :=
[exists H : {group gT}, [&& cyclic H, H <| A & cyclic (A / H)]]. | Definition | metacyclic | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"cyclic",
"gT",
"group"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
metacyclicP A :
reflect (exists H : {group gT}, [/\ cyclic H, H <| A & cyclic (A / H)])
(metacyclic A). | Proof. exact: 'exists_and3P. Qed. | Lemma | metacyclicP | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"cyclic",
"gT",
"group",
"metacyclic"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
metacyclic1 : metacyclic 1. | Proof.
by apply/existsP; exists 1%G; rewrite normal1 trivg_quotient !cyclic1.
Qed. | Lemma | metacyclic1 | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"cyclic1",
"existsP",
"metacyclic",
"normal1",
"trivg_quotient"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cyclic_metacyclic A : cyclic A -> metacyclic A. | Proof.
case/cyclicP=> x ->; apply/existsP; exists (<[x]>)%G.
by rewrite normal_refl cycle_cyclic trivg_quotient cyclic1.
Qed. | Lemma | cyclic_metacyclic | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"cycle_cyclic",
"cyclic",
"cyclic1",
"cyclicP",
"existsP",
"metacyclic",
"normal_refl",
"trivg_quotient"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
metacyclicS G 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 | metacyclicS | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"cyclicS",
"isog_cyclic",
"metacyclic",
"metacyclicP",
"normalGI",
"normal_norm",
"nsKG",
"quotientS",
"sHG",
"second_isog",
"setIC",
"subsetIr",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cyclem & gT | := fun x : gT => x ^+ n. | Definition | cyclem | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"gT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | cyclemM | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"centsP",
"cycle_abelian",
"cyclem",
"expgMn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cyclem_morphism | := Morphism cyclemM. | Canonical | cyclem_morphism | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"cyclemM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | injm_cyclem | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"Zp_cast",
"a1",
"apply",
"cycle1",
"cyclem",
"dvdn1",
"dvdn_gcd",
"eqVneq",
"eqnP",
"inE",
"order_dvdG",
"order_dvdn",
"order_eq1",
"order_gt1",
"setIdP",
"subsetP",
"val",
"valP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | im_cyclem | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"cycleX",
"cycle_id",
"cyclem",
"injm_cyclem",
"morphim_cycle",
"morphim_fixP",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Zp_unitm | := aut injm_cyclem im_cyclem. | Definition | Zp_unitm | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"aut",
"im_cyclem",
"injm_cyclem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 (s... | Lemma | Zp_unitmM | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"Aut_aut",
"Zp_cast",
"Zp_unitm",
"apply",
"autE",
"card_le1_trivg",
"cyclem",
"dvd1n",
"eq_Aut",
"expgM",
"expg_mod_order",
"groupM",
"groupX",
"last",
"leqP",
"modn_dvdm",
"order1",
"order_dvdG",
"permM",
"set1P",
"units_Zp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Zp_unit_morphism | := Morphism Zp_unitmM. | Canonical | Zp_unit_morphism | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"Zp_unitmM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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_... | Lemma | injm_Zp_unitm | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"Zp_cast",
"Zp_unitm",
"a1",
"apply",
"autE",
"card_le1_trivg",
"card_units_Zp",
"cycle_id",
"eqVneq",
"eq_expg_mod_order",
"inE",
"modZp",
"morph1",
"morphpreP",
"order1",
"order_gt1",
"set1P",
"subIset",
"subsetP",
"subxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
generator_coprime m : 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 | generator_coprime | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"coprime",
"cycleX",
"divn1",
"eqEcard",
"eq_sym",
"eqnP",
"eqn_leq",
"gcdn",
"gcdn_gt0",
"generator",
"last",
"leq_divRL",
"leq_pmul2l",
"muln1",
"orderXgcd",
"order_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 ... | Lemma | im_Zp_unitm | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"Aut",
"Aut1",
"Aut_aut",
"Zp_cast",
"Zp_unitm",
"Zpm",
"a1",
"apply",
"autE",
"autmE",
"coprime",
"cycle1",
"cycleP",
"cycle_id",
"cyclem",
"def_n",
"eqVneq",
"eq_Aut",
"expgM",
"generator_coprime",
"im_Zpm",
"im_autm",
"imsetP",
"inE",
"injm_Zpm",
"invm",
"invmK... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Zp_unit_isom : isom (units_Zp #[a]) (Aut <[a]>) Zp_unitm. | Proof. by apply/isomP; rewrite ?injm_Zp_unitm ?im_Zp_unitm. Qed. | Lemma | Zp_unit_isom | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"Aut",
"Zp_unitm",
"apply",
"im_Zp_unitm",
"injm_Zp_unitm",
"isom",
"isomP",
"units_Zp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Zp_unit_isog : isog (units_Zp #[a]) (Aut <[a]>). | Proof. exact: isom_isog Zp_unit_isom. Qed. | Lemma | Zp_unit_isog | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"Aut",
"Zp_unit_isom",
"isog",
"isom_isog",
"units_Zp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_Aut_cycle : #|Aut <[a]>| = totient #[a]. | Proof. by rewrite -(card_isog Zp_unit_isog) card_units_Zp. Qed. | Lemma | card_Aut_cycle | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"Aut",
"Zp_unit_isog",
"card_isog",
"card_units_Zp",
"totient"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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.
by apply/subsetP=> x /[1!inE]; apply: cycle_generator.
rewrite -card_units_Zp // cardsE card_sub morphim_invmE; ap... | Lemma | totient_gen | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"Zp",
"Zp_cast",
"apply",
"card_injm",
"card_le1_trivg",
"card_sub",
"card_units_Zp",
"cards1",
"cardsE",
"cycle_eq1",
"cycle_generator",
"eq_card",
"eq_card1",
"eq_sym",
"generator",
"generator_coprime",
"im_Zpm",
"inE",
"injm_Zpm",
"injm_invm",
"leqP",
"morphim_invmE",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Aut_cycle_abelian : abelian (Aut <[a]>). | Proof. by rewrite -im_Zp_unitm morphim_abelian ?units_Zp_abelian. Qed. | Lemma | Aut_cycle_abelian | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"Aut",
"abelian",
"im_Zp_unitm",
"morphim_abelian",
"units_Zp_abelian"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Aut_cyclic_abelian : cyclic G -> abelian (Aut G). | Proof. by case/cyclicP=> x ->; apply: Aut_cycle_abelian. Qed. | Lemma | Aut_cyclic_abelian | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"Aut",
"Aut_cycle_abelian",
"abelian",
"apply",
"cyclic",
"cyclicP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_Aut_cyclic : cyclic G -> #|Aut G| = totient #|G|. | Proof. by case/cyclicP=> x ->; apply: card_Aut_cycle. Qed. | Lemma | card_Aut_cyclic | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"Aut",
"apply",
"card_Aut_cycle",
"cyclic",
"cyclicP",
"totient"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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).
apply/setP=> x /[!inE]; apply: andb_id2l => Gx.
by rewrite /eq_op /= inordK // ltnS subset_l... | Lemma | sum_ncycle_totient | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"cycle",
"cycle_subG",
"eq_bigr",
"eq_card",
"eq_sym",
"eqxx",
"gT",
"generator",
"imsetP",
"inE",
"inord",
"inordK",
"ltnS",
"order",
"partition_big",
"partition_big_imset",
"setIdP",
"setP",
"subset_leq_card",
"sum1_card",
"sum1dep_card",
"sum_nat_const",
"to... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sum_totient_dvd n : \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]; la... | Lemma | sum_totient_dvd | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"Zp_cycle",
"apply",
"big_mkcond",
"big_ord0",
"big_ord_recl",
"card_Zp",
"cards1",
"cycle_sub_group",
"eq_bigr",
"eq_card0",
"eqxx",
"imsetP",
"imset_f",
"imset_set1",
"inE",
"last",
"mem_cycle",
"mul1n",
"n'",
"order",
"order_dvdG",
"rhs",
"set1P",
"setIdP",
"setP",... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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]]]; ca... | Lemma | order_inj_cyclic | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"addSn",
"apply",
"bigD1",
"big_ind2",
"big_mkcond",
"cardG_gt0",
"cards0",
"cycle_subG",
"cyclic",
"cyclicP",
"dvdnn",
"eqEcard",
"eq_card0",
"eq_card1",
"eq_sym",
"eqxx",
"imsetP",
"imset_f",
"inE",
"last",
"leq_add",
"ltnn",
"mul1n",
"ord_max",
"order_dvdG",
"set... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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... | Lemma | div_ring_mul_group_cyclic | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"abelian",
"all",
"allP",
"apply",
"cardE",
"centsP",
"cycle_subG",
"cyclic",
"diff_roots",
"enum",
"enum_uniq",
"eqEcard",
"eq_mulgV1",
"eq_sym",
"eqxx",
"expgS",
"f1",
"fM",
"gT",
"groupM",
"groupV",
"groupX",
"gsimp",
"hornerE",
"hornerXn",
"inE",
"leqnn",
"l... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 ?g... | Lemma | field_mul_group_cyclic | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"centsP",
"commgP",
"cyclic",
"div_ring_mul_group_cyclic",
"f1",
"fM",
"gT",
"groupM",
"groupV",
"mulKg",
"mulVg",
"mulrCA",
"setD1P",
"subr_eq0",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | field_unit_group_cyclic | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"apply",
"cyclic",
"field_mul_group_cyclic",
"group",
"split",
"unit",
"uval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
units_Zp_cyclic p : prime p -> cyclic (units_Zp p). | Proof. by move/pdiv_id <-; exact: field_unit_group_cyclic. Qed. | Lemma | units_Zp_cyclic | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"cyclic",
"field_unit_group_cyclic",
"pdiv_id",
"prime",
"units_Zp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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... | 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; ex... | Lemma | has_prim_root_subproof | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"Build",
"all",
"apply",
"card_seq_sub",
"cardsT",
"cyclic",
"cyclicP",
"eqn_dvd",
"expgS",
"expr1n",
"exprS",
"field_mul_group_cyclic",
"gT",
"has",
"hasP",
"n_gt0",
"nat",
"order_dvdn",
"prim_expr_order",
"prim_order_exists",
"s0",
"seq",
"seq_sub",
"setT",
"size",
... | This subproof has been extracted out of [has_prim_root] for performance reasons.
See github PR #1059 for further documentation and investigation on this problem. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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_gt... | Lemma | has_prim_root | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"all",
"allP",
"apply",
"eq_sym",
"expr1n",
"exprM",
"exprMn",
"exprSr",
"has",
"has_prim_root_subproof",
"leq_eqVlt",
"ltnNge",
"max_unity_roots",
"mem_unity_roots",
"mul1r",
"mulnC",
"mulrA",
"n_gt0",
"nat",
"prednK",
"seq",
"seq_sub",
"size",
"uniq",
"unity_rootE",... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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... | Lemma | Aut_prime_cycle_cyclic | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"Aut",
"Fp_Zcast",
"apply",
"cast_ord",
"cyclic",
"field_mul_group_cyclic",
"gT",
"im_Zp_unitm",
"injm1",
"injm_Zp_unitm",
"injm_invm",
"invm",
"last",
"morph1",
"morphM",
"prime",
"split",
"val",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | Aut_prime_cyclic | solvable | solvable/cyclic.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"bigop",
"prime",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
"poly",
"GRing.T... | [
"Aut",
"Aut_prime_cycle_cyclic",
"cyclic",
"cyclicP",
"gT",
"group",
"prime",
"prime_cyclic"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"n %:R" | := (n %:R%R). | Notation | n %:R | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
act ij (k : 'Z_p) | := let: (i, j) := ij in (i + k * j, j)%R. | Definition | act | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | actP | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"act",
"addr0",
"addrA",
"apply",
"is_action",
"is_total_action",
"mul0r",
"mulrDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
action | := Action actP. | Canonical | action | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"actP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gactP : is_groupAction [set: 'Z_p * 'Z_p] action. | Proof.
move=> k _ /[1!inE]; apply/andP; split; first by apply/subsetP=> ij _ /[1!inE].
by apply/morphicP=> /= [[i1 j1] [i2 j2] _ _]; rewrite !permE /= mulrDr addrACA.
Qed. | Lemma | gactP | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"action",
"addrACA",
"apply",
"inE",
"is_groupAction",
"morphicP",
"mulrDr",
"permE",
"split",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
groupAction | := GroupAction gactP. | Definition | groupAction | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"gactP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gtype_key : unit. | Proof. by []. Qed. | Fact | gtype_key | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gtype | := locked_with gtype_key (sdprod_groupType groupAction). | Definition | gtype | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"groupAction",
"gtype_key",
"sdprod_groupType"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ngtype | := ncprod [set: gtype]. | Definition | ngtype | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"gtype",
"ncprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ngtypeQ n | := xcprod [set: ngtype 2 n] 'Q_8. | Definition | ngtypeQ | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"ngtype",
"xcprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"p ^{1+2}" | := (Pextraspecial.gtype p) : type_scope. | Notation | p ^{1+2} | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"gtype"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"p ^{1+2}" | := [set: gsort p^{1+2}] : group_scope. | Notation | p ^{1+2} | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"gsort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"p ^{1+2}" | := [set: gsort p^{1+2}]%G : Group_scope. | Notation | p ^{1+2} | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"gsort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"p ^{1+2* n }" | := (Pextraspecial.ngtype p n) : type_scope. | Notation | p ^{1+2* n } | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"ngtype"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"p ^{1+2* n }" | := [set: gsort p^{1+2*n}] : group_scope. | Notation | p ^{1+2* n } | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"gsort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"p ^{1+2* n }" | := [set: gsort p^{1+2*n}]%G : Group_scope. | Notation | p ^{1+2* n } | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"gsort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''D^' n" | := (Pextraspecial.ngtype 2 n) : type_scope. | Notation | ''D^' n | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"ngtype"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''D^' n" | := [set: gsort 'D^n] : group_scope. | Notation | ''D^' n | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"gsort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''D^' n" | := [set: gsort 'D^n]%G : Group_scope. | Notation | ''D^' n | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"gsort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''D^' n * 'Q'" | := (Pextraspecial.ngtypeQ n) : type_scope. | Notation | ''D^' n * 'Q' | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"ngtypeQ"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''D^' n * 'Q'" | := [set: gsort 'D^n*Q] : group_scope. | Notation | ''D^' n * 'Q' | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"gsort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''D^' n * 'Q'" | := [set: gsort 'D^n*Q]%G : Group_scope. | Notation | ''D^' n * 'Q' | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"gsort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
p_gt0 | := ltnW p_gt1. | Let | p_gt0 | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"ltnW",
"p_gt1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gtype | := Pextraspecial.gtype. | Notation | gtype | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
actp | := (Pextraspecial.groupAction p). | Notation | actp | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"groupAction"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | card_pX1p2 | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"Zp_cast",
"card_injm",
"card_ord",
"card_prod",
"cardsT",
"gtype",
"injm_sdpair1",
"injm_sdpair2",
"mulnA",
"sdprod_card",
"sdprod_sdpair"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 -m... | Lemma | Grp_pX1p2 | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"Grp",
"Zp_cast",
"actf",
"actp",
"addr0",
"apply",
"comm1g",
"commXg",
"commgEl",
"commgP",
"commgX",
"commute",
"commuteX",
"commuteX2",
"conjMg",
"conjg",
"cycle_subG",
"eqEsubset",
"existsP",
"expg1n",
"expgD",
"expgM",
"expg_mod",
"f1",
"f2",
"genM_join",
"ge... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pX1p2_pgroup : p.-group p^{1+2}. | Proof. by rewrite /pgroup card_pX1p2 pnatX pnat_id. Qed. | Lemma | pX1p2_pgroup | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"card_pX1p2",
"group",
"pgroup",
"pnatX",
"pnat_id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 //.
re... | Lemma | pX1p2_extraspecial | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"Grp_pX1p2",
"apply",
"card_pX1p2",
"cent_joinEl",
"dvdn_cardMg",
"dvdn_leq_log",
"dvdn_mul",
"existsP",
"expn_gt0",
"extraspecial",
"isoGrp_hom",
"joing_subl",
"joing_subr",
"last",
"leqNgt",
"muln_gt0",
"orderE",
"order_dvdn",
"order_gt0",
"p3group_extraspecial",
"pX1p2_pgr... | This is part of the existence half of Aschbacher ex. (8.7)(1) | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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!eq... | Lemma | exponent_pX1p2 | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"Grp_pX1p2",
"OhmE",
"Ohm_sub",
"apply",
"card_pX1p2",
"eqEsubset",
"eqxx",
"existsP",
"exponent",
"exponent_Ohm1_class2",
"genS",
"inE",
"isoGrp_hom",
"joing_idl",
"joing_idr",
"nil_class2",
"odd",
"oddX",
"pG",
"pX1p2_extraspecial",
"pX1p2_pgroup",
"sub1set",
"subUset",... | This is part of the existence half of Aschbacher ex. (8.7)(1) | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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.
b... | Lemma | isog_pX1p2 | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"Grp_pX1p2",
"Lagrange",
"Phi_quotient_abelem",
"TI_cardMg",
"abelem",
"abelemP",
"apply",
"card_center_extraspecial",
"card_pX1p2",
"card_quotient",
"cent1C",
"cent1P",
"cent1id",
"centP",
"centS",
"cent_cycle",
"cent_joinEr",
"center_normal",
"center_sub",
"commg",
"commgP"... | This is the uniqueness half of Aschbacher ex. (8.7)(1) | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
pX1p2id : p^{1+2*1} \isog p^{1+2}. | Proof. exact: ncprod1. Qed. | Lemma | pX1p2id | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"isog",
"ncprod1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pX1p2S n : xcprod_spec p^{1+2} p^{1+2*n} p^{1+2*n.+1}%type. | Proof. exact: ncprodS. Qed. | Lemma | pX1p2S | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"ncprodS",
"type",
"xcprod_spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_pX1p2n n : 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}... | Lemma | card_pX1p2n | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"cardMg_divn",
"card_center_extraspecial",
"card_injm",
"card_isog",
"card_pX1p2",
"center_sub",
"expnD",
"im_cpair",
"injm_center",
"injm_cpair1g",
"injm_cpairg1",
"isoZ",
"mulKn",
"ncprod0",
"oZ",
"pG",
"pX1p2S",
"pX1p2_extraspecial",
"pX1p2_pgroup",
"p_pr",
"prime",
"pri... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pX1p2n_pgroup n : prime p -> p.-group p^{1+2*n}. | Proof. by move=> p_pr; rewrite /pgroup card_pX1p2n // pnatX pnat_id. Qed. | Lemma | pX1p2n_pgroup | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"card_pX1p2n",
"group",
"p_pr",
"pgroup",
"pnatX",
"pnat_id",
"prime"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exponent_pX1p2n n : 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 ... | Lemma | exponent_pX1p2n | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"apply",
"cardG_gt1",
"card_pX1p2n",
"centsP",
"dvdn1",
"dvdn_trans",
"expgMn",
"exponent",
"exponentP",
"exponent_dvdn",
"exponent_injm",
"exponent_pX1p2",
"im_cpair",
"im_cpair_cent",
"imset2P",
"injm_cpair1g",
"injm_cpairg1",
"isoZ",
"ltn_exp2l",
"mulg1",
"odd",
"pX1p2S"... | This is part of the existence half of Aschbacher (23.13) | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
pX1p2n_extraspecial n : 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: i... | Lemma | pX1p2n_extraspecial | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"apply",
"cprod_extraspecial",
"extraspecial",
"im_cpair_cprod",
"injm_cpair1g",
"injm_cpairg1",
"injm_extraspecial",
"isoZ",
"isog_extraspecial",
"isog_sym",
"n_gt0",
"pX1p2S",
"pX1p2_extraspecial",
"pX1p2id",
"pX1p2n_pgroup",
"p_pr",
"posnP",
"prime",
"setI_im_cpair"
] | This is part of the existence half of Aschbacher (23.13) and (23.14) | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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... | 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... | Lemma | Ohm1_extraspecial_odd | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"Lagrange",
"Lagrange_index",
"Mho",
"MhoS",
"Mho_p_elt",
"OhmE",
"OhmS",
"Ohm_normal",
"Ohm_sub",
"Phi_joing",
"Phi_quotient_abelem",
"TI_Ohm1",
"TI_cardMg",
"Uu",
"abelem",
"abelemS",
"abelian",
"abelianE",
"apply",
"cardG_gt0",
"cardG_gt1",
"cardSg",
"card_center_extra... | This is Aschbacher (23.12) | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
isog_pX1p2n n (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.
e... | Lemma | isog_pX1p2n | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"Aut_extraspecial_full",
"abelian",
"apply",
"big_cons",
"big_nil",
"cardG_gt0",
"card_center_extraspecial",
"card_p3group_extraspecial",
"card_pX1p2n",
"center_idP",
"centsC",
"cprod1g",
"cprodA",
"cprodP",
"defG",
"def_n",
"double_inj",
"dvdn_trans",
"eqEsubset",
"eqSS",
"e... | in part the proof that symplectic spaces are hyperbolic (19.16). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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 rewri... | Lemma | isog_2X1p2 | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"Grp_pX1p2",
"apply",
"card_pX1p2",
"cycle1",
"defG",
"existsP",
"involutions_gen_dihedral",
"isoGrp_hom",
"isog",
"joing1G",
"joingG1",
"joing_idPl",
"joing_idl",
"joing_idr",
"nt_prime_order",
"orderE",
"order_dvdn",
"prime"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | Q8_extraspecial | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"apply",
"extraspecial",
"generators_quaternion",
"isog",
"isog_refl",
"last",
"oZ",
"quaternion_structure",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
DnQ_P n : 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... | Lemma | DnQ_P | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"Q8_extraspecial",
"apply",
"card_center_extraspecial",
"card_pX1p2n",
"card_quaternion",
"center_ncprod0",
"group",
"isog_cyclic_card",
"n_gt0",
"pX1p2n_extraspecial",
"pX1p2n_pgroup",
"pgroup",
"posnP",
"prime",
"prime_cyclic",
"type",
"xcprodP",
"xcprod_spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_DnQ n : #|'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//; first exact: injm_cpair1g.
rewrite (card_injm (injm_cpairg1 _))//= (card_injm (injm_cpair1g _))//.
rewr... | Lemma | card_DnQ | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"DnQ_P",
"Q8_extraspecial",
"cardMg_divn",
"card_center_extraspecial",
"card_injm",
"card_pX1p2n",
"card_quaternion",
"center_sub",
"cpair_center_id",
"expnD",
"group",
"im_cpair",
"injm_center",
"injm_cpair1g",
"injm_cpairg1",
"isoZ",
"mulnC",
"muln_divA",
"pgroup",
"setI_im_c... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
DnQ_pgroup n : 2.-group 'D^n*Q. | Proof. by rewrite /pgroup card_DnQ pnatX. Qed. | Lemma | DnQ_pgroup | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"card_DnQ",
"group",
"pgroup",
"pnatX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
DnQ_extraspecial n : 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; last exact: injm_extraspecial Q8_extraspecial.
apply/setIidPl; rewrite setI_im_cpair -injm_center //=.
by congr (_ @* _); rewrite n0 center_ncpro... | Lemma | DnQ_extraspecial | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"DnQ_P",
"DnQ_pgroup",
"Q8_extraspecial",
"apply",
"center_ncprod0",
"cprod_extraspecial",
"extraspecial",
"im_cpair",
"im_cpair_cprod",
"injm_center",
"injm_cpair1g",
"injm_cpairg1",
"injm_extraspecial",
"isoZ",
"last",
"mulSGid",
"n_gt0",
"pX1p2n_extraspecial",
"posnP",
"setI... | Final part of the existence half of Aschbacher (23.14). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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_cycli... | Lemma | card_isog8_extraspecial | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"apply",
"cycle_cyclic",
"cycle_subG",
"cyclic",
"divgS",
"exponent_2extraspecial",
"exponent_witness",
"extraspecial",
"extraspecial_nonabelian",
"extremal2",
"extremal_class",
"gT",
"group",
"isog",
"maximal_cycle_extremal",
"orderE",
"pG",
"pgroup",
"pgroup_nil",
"sXG"
] | A special case of the uniqueness half of Achsbacher (23.14). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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 rewrit... | Lemma | isog_2extraspecial | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"Aut",
"Aut_extraspecial_full",
"Aut_in",
"DnQ_P",
"LdivP",
"OhmE",
"apply",
"card_center_extraspecial",
"card_isog8_extraspecial",
"center_ncprod0",
"center_prod",
"center_sub",
"cpair1g",
"cpairg1",
"cpairg1_center",
"cprodA",
"cprodE",
"cprodP",
"cprod_center_id",
"cycle_sub... | Galois theory as in (20.9) and (21.1). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
rank_Dn n : '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 /... | Lemma | rank_Dn | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"Dx",
"Lagrange",
"abelE",
"apply",
"cardG_gt0",
"card_center_extraspecial",
"card_pX1p2n",
"card_subcent_extraspecial",
"centSS",
"cpair1g",
"cpairg1",
"cycleX",
"cycle_subG",
"dihedral2_structure",
"doubleS",
"dprodEY",
"dprod_abelem",
"dprod_card",
"eqn_leq",
"expnM",
"exp... | The first concluding remark of Aschbacher (23.14). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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