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 |
|---|---|---|---|---|---|---|
enum_valP:= enum_valP.
|
Notation
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
enum_valP
| |
enum_val_nth:= enum_val_nth.
|
Notation
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
enum_val_nth
| |
nth_enum_rank_in:= nth_enum_rank_in.
|
Notation
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
nth_enum_rank_in
| |
nth_enum_rank:= nth_enum_rank.
|
Notation
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
nth_enum_rank
| |
enum_rankK_in:= enum_rankK_in.
|
Notation
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
enum_rankK_in
| |
enum_rankK:= enum_rankK.
|
Notation
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
enum_rankK
| |
enum_valK_in:= enum_valK_in.
|
Notation
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
enum_valK_in
| |
enum_valK:= enum_valK.
|
Notation
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
enum_valK
| |
enum_rank_inj:= enum_rank_inj.
|
Notation
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
enum_rank_inj
| |
enum_val_inj:= enum_val_inj.
|
Notation
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
enum_val_inj
| |
enum_val_bij_in:= enum_val_bij_in.
|
Notation
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
enum_val_bij_in
| |
eq_enum_rank_in:= eq_enum_rank_in.
|
Notation
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
eq_enum_rank_in
| |
enum_rank_in_inj:= enum_rank_in_inj.
|
Notation
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
enum_rank_in_inj
| |
enum_rank_bij:= enum_rank_bij.
|
Notation
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
enum_rank_bij
| |
enum_val_bij:= enum_val_bij.
|
Notation
|
order
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import path fintype tuple bigop finset div prime finfun",
"From mathcomp Require Import finset"
] |
order/preorder.v
|
enum_val_bij
| |
Ldivn := [set x : gT | x ^+ n == 1].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
Ldiv
| |
exponentA := \big[lcmn/1%N]_(x in A) #[x].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
exponent
| |
abelemp A := [&& p.-group A, abelian A & exponent A %| p].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
abelem
| |
is_abelemA := abelem (pdiv #|A|) A.
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
is_abelem
| |
pElemp A := [set E : {group gT} | E \subset A & abelem p E].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
pElem
| |
pnElemp n A := [set E in pElem p A | logn p #|E| == n].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
pnElem
| |
nElemn A := \bigcup_(0 <= p < #|A|.+1) pnElem p n A.
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
nElem
| |
pmaxElemp A := [set E | [max E | E \in pElem p A]].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
pmaxElem
| |
p_rankp A := \max_(E in pElem p A) logn p #|E|.
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
p_rank
| |
rankA := \max_(0 <= p < #|A|.+1) p_rank p A.
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
rank
| |
gen_rankA := #|[arg min_(B < A | <<B>> == A) #|B|]|.
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
gen_rank
| |
Ohm:= <<[set x in A | x ^+ (pdiv #[x] ^ n) == 1]>>.
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
Ohm
| |
Mho:= <<[set x ^+ (pdiv #[x] ^ n) | x in A & (pdiv #[x]).-elt x]>>.
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
Mho
| |
Ohm_group: {group gT} := Eval hnf in [group of Ohm].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
Ohm_group
| |
Mho_group: {group gT} := Eval hnf in [group of Mho].
|
Canonical
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
Mho_group
| |
pdiv_p_elt(p : nat) (x : gT) : p.-elt x -> x != 1 -> pdiv #[x] = p.
Proof.
move=> p_x; rewrite /order -cycle_eq1.
by case/(pgroup_pdiv p_x)=> p_pr _ [k ->]; rewrite pdiv_pfactor.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
pdiv_p_elt
| |
OhmPredP(x : gT) :
reflect (exists2 p, prime p & x ^+ (p ^ n) = 1) (x ^+ (pdiv #[x] ^ n) == 1).
Proof.
have [-> | nt_x] := eqVneq x 1.
by rewrite expg1n eqxx; left; exists 2; rewrite ?expg1n.
apply: (iffP idP) => [/eqP | [p p_pr /eqP x_pn]].
by exists (pdiv #[x]); rewrite ?pdiv_prime ?order_gt1.
rewrite (@pdiv_p_elt p) //; rewrite -order_dvdn in x_pn.
by rewrite [p_elt _ _](pnat_dvd x_pn) // pnatX pnat_id.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
OhmPredP
| |
Mho_p_elt(p : nat) x : x \in A -> p.-elt x -> x ^+ (p ^ n) \in Mho.
Proof.
move=> Ax p_x; have [-> | ntx] := eqVneq x 1; first by rewrite groupX.
by apply/mem_gen/imsetP; exists x; rewrite ?inE ?Ax (pdiv_p_elt p_x).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
Mho_p_elt
| |
LdivPA n x : reflect (x \in A /\ x ^+ n = 1) (x \in 'Ldiv_n(A)).
Proof. by rewrite !inE; apply: (iffP andP) => [] [-> /eqP]. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
LdivP
| |
dvdn_exponentx A : x \in A -> #[x] %| exponent A.
Proof. by move=> Ax; rewrite (biglcmn_sup x). Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
dvdn_exponent
| |
expg_exponentx A : x \in A -> x ^+ exponent A = 1.
Proof. by move=> Ax; apply/eqP; rewrite -order_dvdn dvdn_exponent. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
expg_exponent
| |
exponentSA B : A \subset B -> exponent A %| exponent B.
Proof.
by move=> sAB; apply/dvdn_biglcmP=> x Ax; rewrite dvdn_exponent ?(subsetP sAB).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
exponentS
| |
exponentPA n :
reflect (forall x, x \in A -> x ^+ n = 1) (exponent A %| n).
Proof.
apply: (iffP (dvdn_biglcmP _ _ _)) => eAn x Ax.
by apply/eqP; rewrite -order_dvdn eAn.
by rewrite order_dvdn eAn.
Qed.
Arguments exponentP {A n}.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
exponentP
| |
trivg_exponentG : (G :==: 1) = (exponent G %| 1).
Proof.
rewrite -subG1.
by apply/subsetP/exponentP=> trG x /trG; rewrite expg1 => /set1P.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
trivg_exponent
| |
exponent1: exponent [1 gT] = 1%N.
Proof. by apply/eqP; rewrite -dvdn1 -trivg_exponent eqxx. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
exponent1
| |
exponent_dvdnG : exponent G %| #|G|.
Proof. by apply/dvdn_biglcmP=> x Gx; apply: order_dvdG. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
exponent_dvdn
| |
exponent_gt0G : 0 < exponent G.
Proof. exact: dvdn_gt0 (exponent_dvdn G). Qed.
Hint Resolve exponent_gt0 : core.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
exponent_gt0
| |
pnat_exponentpi G : pi.-nat (exponent G) = pi.-group G.
Proof.
congr (_ && _); first by rewrite cardG_gt0 exponent_gt0.
apply: eq_all_r => p; rewrite !mem_primes cardG_gt0 exponent_gt0 /=.
apply: andb_id2l => p_pr; apply/idP/idP=> pG.
exact: dvdn_trans pG (exponent_dvdn G).
by case/Cauchy: pG => // x Gx <-; apply: dvdn_exponent.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
pnat_exponent
| |
exponentJA x : exponent (A :^ x) = exponent A.
Proof.
rewrite /exponent (reindex_inj (conjg_inj x)).
by apply: eq_big => [y | y _]; rewrite ?orderJ ?memJ_conjg.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
exponentJ
| |
exponent_witnessG : nilpotent G -> {x | x \in G & exponent G = #[x]}.
Proof.
move=> nilG; have [//=| /= x Gx max_x] := @arg_maxnP _ 1 [in G] order.
exists x => //; apply/eqP; rewrite eqn_dvd dvdn_exponent // andbT.
apply/dvdn_biglcmP=> y Gy; apply/dvdn_partP=> //= p.
rewrite mem_primes => /andP[p_pr _]; have p_gt1: p > 1 := prime_gt1 p_pr.
rewrite p_part pfactor_dvdn // -(leq_exp2l _ _ p_gt1) -!p_part.
rewrite -(leq_pmul2r (part_gt0 p^' #[x])) partnC // -!order_constt.
rewrite -orderM ?order_constt ?coprime_partC // ?max_x ?groupM ?groupX //.
case/dprodP: (nilpotent_pcoreC p nilG) => _ _ cGpGp' _.
have inGp := mem_normal_Hall (nilpotent_pcore_Hall _ nilG) (pcore_normal _ _).
by red; rewrite -(centsP cGpGp') // inGp ?p_elt_constt ?groupX.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
exponent_witness
| |
exponent_cyclex : exponent <[x]> = #[x].
Proof. by apply/eqP; rewrite eqn_dvd exponent_dvdn dvdn_exponent ?cycle_id. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
exponent_cycle
| |
exponent_cyclicX : cyclic X -> exponent X = #|X|.
Proof. by case/cyclicP=> x ->; apply: exponent_cycle. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
exponent_cyclic
| |
primes_exponentG : primes (exponent G) = primes (#|G|).
Proof.
apply/eq_primes => p; rewrite !mem_primes exponent_gt0 cardG_gt0 /=.
by apply: andb_id2l => p_pr; apply: negb_inj; rewrite -!p'natE // pnat_exponent.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
primes_exponent
| |
pi_of_exponentG : \pi(exponent G) = \pi(G).
Proof. by rewrite /pi_of primes_exponent. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
pi_of_exponent
| |
partn_exponentSpi H G :
H \subset G -> #|G|`_pi %| #|H| -> ((exponent H)`_pi = (exponent G)`_pi)%N.
Proof.
move=> sHG Gpi_dvd_H; apply/eqP; rewrite eqn_dvd.
rewrite partn_dvd ?exponentS ?exponent_gt0 //=; apply/dvdn_partP=> // p.
rewrite pi_of_part ?exponent_gt0 // => /andP[_ /= pi_p].
have sppi: {subset (p : nat_pred) <= pi} by move=> q /eqnP->.
have [P sylP] := Sylow_exists p H; have sPH := pHall_sub sylP.
have{} sylP: p.-Sylow(G) P.
rewrite pHallE (subset_trans sPH) //= (card_Hall sylP) eqn_dvd andbC.
by rewrite -{1}(partn_part _ sppi) !partn_dvd ?cardSg ?cardG_gt0.
rewrite partn_part ?partn_biglcm //.
apply: (@big_ind _ (dvdn^~ _)) => [|m n|x Gx]; first exact: dvd1n.
by rewrite dvdn_lcm => ->.
rewrite -order_constt; have p_y := p_elt_constt p x; set y := x.`_p in p_y *.
have sYG: <[y]> \subset G by rewrite cycle_subG groupX.
have [z _ Pyz] := Sylow_Jsub sylP sYG p_y.
rewrite (bigD1 (y ^ z)) ?(subsetP sPH) -?cycle_subG ?cycleJ //=.
by rewrite orderJ part_pnat_id ?dvdn_lcml // (pi_pnat p_y).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
partn_exponentS
| |
exponent_Hallpi G H : pi.-Hall(G) H -> exponent H = ((exponent G)`_pi)%N.
Proof.
move=> hallH; have [sHG piH _] := and3P hallH.
rewrite -(partn_exponentS sHG) -?(card_Hall hallH) ?part_pnat_id //.
by apply: pnat_dvd piH; apply: exponent_dvdn.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
exponent_Hall
| |
exponent_ZgroupG : Zgroup G -> exponent G = #|G|.
Proof.
move/forall_inP=> ZgG; apply/eqP; rewrite eqn_dvd exponent_dvdn.
apply/(dvdn_partP _ (cardG_gt0 _)) => p _.
have [S sylS] := Sylow_exists p G; rewrite -(card_Hall sylS).
have /cyclicP[x defS]: cyclic S by rewrite ZgG ?(p_Sylow sylS).
by rewrite defS dvdn_exponent // -cycle_subG -defS (pHall_sub sylS).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
exponent_Zgroup
| |
cprod_exponentA B G :
A \* B = G -> lcmn (exponent A) (exponent B) = (exponent G).
Proof.
case/cprodP=> [[K H -> ->{A B}] <- cKH].
apply/eqP; rewrite eqn_dvd dvdn_lcm !exponentS ?mulG_subl ?mulG_subr //=.
apply/exponentP=> _ /imset2P[x y Kx Hy ->].
rewrite -[1]mulg1 expgMn; last by red; rewrite -(centsP cKH).
congr (_ * _); apply/eqP; rewrite -order_dvdn.
by rewrite (dvdn_trans (dvdn_exponent Kx)) ?dvdn_lcml.
by rewrite (dvdn_trans (dvdn_exponent Hy)) ?dvdn_lcmr.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
cprod_exponent
| |
dprod_exponentA B G :
A \x B = G -> lcmn (exponent A) (exponent B) = (exponent G).
Proof.
case/dprodP=> [[K H -> ->{A B}] defG cKH _].
by apply: cprod_exponent; rewrite cprodE.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
dprod_exponent
| |
sub_LdivTA n : (A \subset 'Ldiv_n()) = (exponent A %| n).
Proof. by apply/subsetP/exponentP=> eAn x /eAn /[1!inE] /eqP. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
sub_LdivT
| |
LdivT_Jn x : 'Ldiv_n() :^ x = 'Ldiv_n().
Proof.
apply/setP=> y; rewrite !inE mem_conjg inE -conjXg.
by rewrite (canF_eq (conjgKV x)) conj1g.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
LdivT_J
| |
LdivJn A x : 'Ldiv_n(A :^ x) = 'Ldiv_n(A) :^ x.
Proof. by rewrite conjIg LdivT_J. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
LdivJ
| |
sub_LdivA n : (A \subset 'Ldiv_n(A)) = (exponent A %| n).
Proof. by rewrite subsetI subxx sub_LdivT. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
sub_Ldiv
| |
group_LdivG n : abelian G -> group_set 'Ldiv_n(G).
Proof.
move=> cGG; apply/group_setP.
split=> [|x y]; rewrite !inE ?group1 ?expg1n //=.
case/andP=> Gx /eqP xn /andP[Gy /eqP yn].
by rewrite groupM //= expgMn ?xn ?yn ?mulg1 //; apply: (centsP cGG).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
group_Ldiv
| |
abelian_exponent_genA : abelian A -> exponent <<A>> = exponent A.
Proof.
rewrite -abelian_gen; set n := exponent A; set G := <<A>> => cGG.
apply/eqP; rewrite eqn_dvd andbC exponentS ?subset_gen //= -sub_Ldiv.
rewrite -(gen_set_id (group_Ldiv n cGG)) genS // subsetI subset_gen /=.
by rewrite sub_LdivT.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
abelian_exponent_gen
| |
abelem_pgroupp A : p.-abelem A -> p.-group A.
Proof. by case/andP. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
abelem_pgroup
| |
abelem_abelianp A : p.-abelem A -> abelian A.
Proof. by case/and3P. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
abelem_abelian
| |
abelem1p : p.-abelem [1 gT].
Proof. by rewrite /abelem pgroup1 abelian1 exponent1 dvd1n. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
abelem1
| |
abelemEp G : prime p -> p.-abelem G = abelian G && (exponent G %| p).
Proof.
move=> p_pr; rewrite /abelem -pnat_exponent andbA -!(andbC (_ %| _)).
by case: (dvdn_pfactor _ 1 p_pr) => // [[k _ ->]]; rewrite pnatX pnat_id.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
abelemE
| |
abelemPp G :
prime p ->
reflect (abelian G /\ forall x, x \in G -> x ^+ p = 1) (p.-abelem G).
Proof.
by move=> p_pr; rewrite abelemE //; apply: (iffP andP) => [] [-> /exponentP].
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
abelemP
| |
abelem_order_pp G x : p.-abelem G -> x \in G -> x != 1 -> #[x] = p.
Proof.
case/and3P=> pG _ eG Gx; rewrite -cycle_eq1 => ntX.
have{ntX} [p_pr p_x _] := pgroup_pdiv (mem_p_elt pG Gx) ntX.
by apply/eqP; rewrite eqn_dvd p_x andbT order_dvdn (exponentP eG).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
abelem_order_p
| |
cyclic_abelem_primep X : p.-abelem X -> cyclic X -> X :!=: 1 -> #|X| = p.
Proof.
move=> abelX cycX; case/cyclicP: cycX => x -> in abelX *.
by rewrite cycle_eq1; apply: abelem_order_p abelX (cycle_id x).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
cyclic_abelem_prime
| |
cycle_abelemp x : p.-elt x || prime p -> p.-abelem <[x]> = (#[x] %| p).
Proof.
move=> p_xVpr; rewrite /abelem cycle_abelian /=.
apply/andP/idP=> [[_ xp1] | x_dvd_p].
by rewrite order_dvdn (exponentP xp1) ?cycle_id.
split; last exact: dvdn_trans (exponent_dvdn _) x_dvd_p.
by case/orP: p_xVpr => // /pnat_id; apply: pnat_dvd.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
cycle_abelem
| |
exponent2_abelemG : exponent G %| 2 -> 2.-abelem G.
Proof.
move/exponentP=> expG; apply/abelemP=> //; split=> //.
apply/centsP=> x Gx y Gy; apply: (mulIg x); apply: (mulgI y).
by rewrite -!mulgA !(mulgA y) -!(expgS _ 1) !expG ?mulg1 ?groupM.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
exponent2_abelem
| |
prime_abelemp G : prime p -> #|G| = p -> p.-abelem G.
Proof.
move=> p_pr oG; rewrite /abelem -oG exponent_dvdn.
by rewrite /pgroup cyclic_abelian ?prime_cyclic ?oG ?pnat_id.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
prime_abelem
| |
abelem_cyclicp G : p.-abelem G -> cyclic G = (logn p #|G| <= 1).
Proof.
move=> abelG; have [pG _ expGp] := and3P abelG.
case: (eqsVneq G 1) => [-> | ntG]; first by rewrite cyclic1 cards1 logn1.
have [p_pr _ [e oG]] := pgroup_pdiv pG ntG; apply/idP/idP.
case/cyclicP=> x defG; rewrite -(pfactorK 1 p_pr) dvdn_leq_log ?prime_gt0 //.
by rewrite defG order_dvdn (exponentP expGp) // defG cycle_id.
by rewrite oG pfactorK // ltnS leqn0 => e0; rewrite prime_cyclic // oG (eqP e0).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
abelem_cyclic
| |
abelemSp H G : H \subset G -> p.-abelem G -> p.-abelem H.
Proof.
move=> sHG /and3P[cGG pG Gp1]; rewrite /abelem.
by rewrite (pgroupS sHG) // (abelianS sHG) // (dvdn_trans (exponentS sHG)).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
abelemS
| |
abelemJp G x : p.-abelem (G :^ x) = p.-abelem G.
Proof. by rewrite /abelem pgroupJ abelianJ exponentJ. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
abelemJ
| |
cprod_abelemp A B G :
A \* B = G -> p.-abelem G = p.-abelem A && p.-abelem B.
Proof.
case/cprodP=> [[H K -> ->{A B}] defG cHK].
apply/idP/andP=> [abelG | []].
by rewrite !(abelemS _ abelG) // -defG (mulG_subl, mulG_subr).
case/and3P=> pH cHH expHp; case/and3P=> pK cKK expKp.
rewrite -defG /abelem pgroupM pH pK abelianM cHH cKK cHK /=.
apply/exponentP=> _ /imset2P[x y Hx Ky ->].
rewrite expgMn; last by red; rewrite -(centsP cHK).
by rewrite (exponentP expHp) // (exponentP expKp) // mul1g.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
cprod_abelem
| |
dprod_abelemp A B G :
A \x B = G -> p.-abelem G = p.-abelem A && p.-abelem B.
Proof.
move=> defG; case/dprodP: (defG) => _ _ _ tiHK.
by apply: cprod_abelem; rewrite -dprodEcp.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
dprod_abelem
| |
is_abelem_pgroupp G : p.-group G -> is_abelem G = p.-abelem G.
Proof.
rewrite /is_abelem => pG.
case: (eqsVneq G 1) => [-> | ntG]; first by rewrite !abelem1.
by have [p_pr _ [k ->]] := pgroup_pdiv pG ntG; rewrite pdiv_pfactor.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
is_abelem_pgroup
| |
is_abelemPG : reflect (exists2 p, prime p & p.-abelem G) (is_abelem G).
Proof.
apply: (iffP idP) => [abelG | [p p_pr abelG]].
case: (eqsVneq G 1) => [-> | ntG]; first by exists 2; rewrite ?abelem1.
by exists (pdiv #|G|); rewrite ?pdiv_prime // ltnNge -trivg_card_le1.
by rewrite (is_abelem_pgroup (abelem_pgroup abelG)).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
is_abelemP
| |
pElemPp A E : reflect (E \subset A /\ p.-abelem E) (E \in 'E_p(A)).
Proof. by rewrite inE; apply: andP. Qed.
Arguments pElemP {p A E}.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
pElemP
| |
pElemSp A B : A \subset B -> 'E_p(A) \subset 'E_p(B).
Proof.
by move=> sAB; apply/subsetP=> E /[!inE] /andP[/subset_trans->].
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
pElemS
| |
pElemIp A B : 'E_p(A :&: B) = 'E_p(A) :&: subgroups B.
Proof. by apply/setP=> E; rewrite !inE subsetI andbAC. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
pElemI
| |
pElemJx p A E : ((E :^ x)%G \in 'E_p(A :^ x)) = (E \in 'E_p(A)).
Proof. by rewrite !inE conjSg abelemJ. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
pElemJ
| |
pnElemPp n A E :
reflect [/\ E \subset A, p.-abelem E & logn p #|E| = n] (E \in 'E_p^n(A)).
Proof. by rewrite !inE -andbA; apply: (iffP and3P) => [] [-> -> /eqP]. Qed.
Arguments pnElemP {p n A E}.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
pnElemP
| |
pnElemPcardp n A E :
E \in 'E_p^n(A) -> [/\ E \subset A, p.-abelem E & #|E| = p ^ n]%N.
Proof.
by case/pnElemP=> -> abelE <-; rewrite -card_pgroup // abelem_pgroup.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
pnElemPcard
| |
card_pnElemp n A E : E \in 'E_p^n(A) -> #|E| = (p ^ n)%N.
Proof. by case/pnElemPcard. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
card_pnElem
| |
pnElem0p G : 'E_p^0(G) = [set 1%G].
Proof.
apply/setP=> E; rewrite !inE -andbA; apply/and3P/idP=> [[_ pE] | /eqP->].
apply: contraLR; case/(pgroup_pdiv (abelem_pgroup pE)) => p_pr _ [k ->].
by rewrite pfactorK.
by rewrite sub1G abelem1 cards1 logn1.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
pnElem0
| |
pnElem_primep n A E : E \in 'E_p^n.+1(A) -> prime p.
Proof. by case/pnElemP=> _ _; rewrite lognE; case: prime. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
pnElem_prime
| |
pnElemEp n A :
prime p -> 'E_p^n(A) = [set E in 'E_p(A) | #|E| == (p ^ n)%N].
Proof.
move/pfactorK=> pnK; apply/setP=> E; rewrite 3!inE.
case: (@andP (E \subset A)) => //= [[_]] /andP[/p_natP[k ->] _].
by rewrite pnK (can_eq pnK).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
pnElemE
| |
pnElemSp n A B : A \subset B -> 'E_p^n(A) \subset 'E_p^n(B).
Proof.
move=> sAB; apply/subsetP=> E.
by rewrite !inE -!andbA => /andP[/subset_trans->].
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
pnElemS
| |
pnElemIp n A B : 'E_p^n(A :&: B) = 'E_p^n(A) :&: subgroups B.
Proof. by apply/setP=> E; rewrite !inE subsetI -!andbA; do !bool_congr. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
pnElemI
| |
pnElemJx p n A E : ((E :^ x)%G \in 'E_p^n(A :^ x)) = (E \in 'E_p^n(A)).
Proof. by rewrite inE pElemJ cardJg !inE. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
pnElemJ
| |
abelem_pnElemp n G :
p.-abelem G -> n <= logn p #|G| -> exists E, E \in 'E_p^n(G).
Proof.
case: n => [|n] abelG lt_nG; first by exists 1%G; rewrite pnElem0 set11.
have p_pr: prime p by move: lt_nG; rewrite lognE; case: prime.
case/(normal_pgroup (abelem_pgroup abelG)): lt_nG => // E [sEG _ oE].
by exists E; rewrite pnElemE // !inE oE sEG (abelemS sEG) /=.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
abelem_pnElem
| |
card_p1Elemp A X : X \in 'E_p^1(A) -> #|X| = p.
Proof. exact: card_pnElem. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
card_p1Elem
| |
p1ElemEp A : prime p -> 'E_p^1(A) = [set X in subgroups A | #|X| == p].
Proof.
move=> p_pr; apply/setP=> X; rewrite pnElemE // !inE -andbA; congr (_ && _).
by apply: andb_idl => /eqP oX; rewrite prime_abelem ?oX.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
p1ElemE
| |
TIp1ElemPp A X Y :
X \in 'E_p^1(A) -> Y \in 'E_p^1(A) -> reflect (X :&: Y = 1) (X :!=: Y).
Proof.
move=> EpX EpY; have p_pr := pnElem_prime EpX.
have [oX oY] := (card_p1Elem EpX, card_p1Elem EpY).
have [<-|] := eqVneq.
by right=> X1; rewrite -oX -(setIid X) X1 cards1 in p_pr.
by rewrite eqEcard oX oY leqnn andbT; left; rewrite prime_TIg ?oX.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
TIp1ElemP
| |
card_p1Elem_pnElemp n A E :
E \in 'E_p^n(A) -> #|'E_p^1(E)| = (\sum_(i < n) p ^ i)%N.
Proof.
case/pnElemP=> _ {A} abelE dimE; have [pE cEE _] := and3P abelE.
have [E1 | ntE] := eqsVneq E 1.
rewrite -dimE E1 cards1 logn1 big_ord0 eq_card0 // => X.
by rewrite !inE subG1 trivg_card1; case: eqP => // ->; rewrite logn1 andbF.
have [p_pr _ _] := pgroup_pdiv pE ntE; have p_gt1 := prime_gt1 p_pr.
apply/eqP; rewrite -(@eqn_pmul2l (p - 1)) ?subn_gt0 // subn1 -predn_exp.
have groupD1_inj: injective (fun X => (gval X)^#).
apply: can_inj (@generated_group _) _ => X.
by apply: val_inj; rewrite /= genD1 ?group1 ?genGid.
rewrite -dimE -card_pgroup // (cardsD1 1 E) group1 /= mulnC.
rewrite -(card_imset _ groupD1_inj) eq_sym.
apply/eqP; apply: card_uniform_partition => [X'|].
case/imsetP=> X; rewrite pnElemE // expn1 => /setIdP[_ /eqP <-] ->.
by rewrite (cardsD1 1 X) group1.
apply/and3P; split; last 1 first.
- apply/imsetP=> [[X /card_p1Elem oX X'0]].
by rewrite -oX (cardsD1 1) -X'0 group1 cards0 in p_pr.
- rewrite eqEsubset; apply/andP; split.
by apply/bigcupsP=> _ /imsetP[X /pnElemP[sXE _ _] ->]; apply: setSD.
apply/subsetP=> x /setD1P[ntx Ex].
apply/bigcupP; exists <[x]>^#; last by rewrite !inE ntx cycle_id.
apply/imsetP; exists <[x]>%G; rewrite ?p1ElemE // !inE cycle_subG Ex /=.
by rewrite -orderE (abelem_order_p abelE).
apply/trivIsetP=> _ _ /imsetP[X EpX ->] /imsetP[Y EpY ->]; apply/implyP.
rewrite (inj_eq groupD1_inj) -setI_eq0 -setDIl setD_eq0 subG1.
by rewrite (sameP eqP (TI
...
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
card_p1Elem_pnElem
| |
card_p1Elem_p2Elemp A E : E \in 'E_p^2(A) -> #|'E_p^1(E)| = p.+1.
Proof. by move/card_p1Elem_pnElem->; rewrite big_ord_recl big_ord1. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
card_p1Elem_p2Elem
| |
p2Elem_dprodPp A E X Y :
E \in 'E_p^2(A) -> X \in 'E_p^1(E) -> Y \in 'E_p^1(E) ->
reflect (X \x Y = E) (X :!=: Y).
Proof.
move=> Ep2E EpX EpY; have [_ abelE oE] := pnElemPcard Ep2E.
apply: (iffP (TIp1ElemP EpX EpY)) => [tiXY|]; last by case/dprodP.
have [[sXE _ oX] [sYE _ oY]] := (pnElemPcard EpX, pnElemPcard EpY).
rewrite dprodE ?(sub_abelian_cent2 (abelem_abelian abelE)) //.
by apply/eqP; rewrite eqEcard mul_subG //= TI_cardMg // oX oY oE.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
p2Elem_dprodP
| |
nElemPn G E : reflect (exists p, E \in 'E_p^n(G)) (E \in 'E^n(G)).
Proof.
rewrite ['E^n(G)]big_mkord.
apply: (iffP bigcupP) => [[[p /= _] _] | [p]]; first by exists p.
case: n => [|n EpnE]; first by rewrite pnElem0; exists ord0; rewrite ?pnElem0.
suffices lepG: p < #|G|.+1 by exists (Ordinal lepG).
have:= EpnE; rewrite pnElemE ?(pnElem_prime EpnE) // !inE -andbA ltnS.
case/and3P=> sEG _ oE; rewrite dvdn_leq // (dvdn_trans _ (cardSg sEG)) //.
by rewrite (eqP oE) dvdn_exp.
Qed.
Arguments nElemP {n G E}.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
nElemP
| |
nElem0G : 'E^0(G) = [set 1%G].
Proof.
apply/setP=> E; apply/nElemP/idP=> [[p] |]; first by rewrite pnElem0.
by exists 2; rewrite pnElem0.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
nElem0
| |
nElem1PG E :
reflect (E \subset G /\ exists2 p, prime p & #|E| = p) (E \in 'E^1(G)).
Proof.
apply: (iffP nElemP) => [[p pE] | [sEG [p p_pr oE]]].
have p_pr := pnElem_prime pE; rewrite pnElemE // !inE -andbA in pE.
by case/and3P: pE => -> _ /eqP; split; last exists p.
exists p; rewrite pnElemE // !inE sEG oE eqxx abelemE // -oE exponent_dvdn.
by rewrite cyclic_abelian // prime_cyclic // oE.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import choice div fintype finfun bigop finset prime",
"From mathcomp Require Import binomial fingroup morphism perm automorphism",
"From mathcomp Require Import action quotient gfunctor gproduct ssralg countalg",
"From mathcomp Require Import finalg zmodp cyclic pgroup gseries nilpotent sylow"
] |
solvable/abelian.v
|
nElem1P
|
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