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 |
|---|---|---|---|---|---|---|
Grp_modular_group:
'Mod_(p ^ n) \isog Grp (x : y : x ^+ q, y ^+ p, x ^ y = x ^+ r.+1).
Proof.
rewrite /modular_gtype def_p def_q def_r; apply: Extremal.Grp => //.
set B := <[_]>; have Bb: Zp1 \in B by apply: cycle_id.
have oB: #|B| = q by rewrite -orderE order_Zp1 Zp_cast.
have cycB: cyclic B by rewrite cycle_cyclic.... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | Grp_modular_group | |
modular_group_generatorsgT (xy : gT * gT) :=
let: (x, y) := xy in #[y] = p /\ x ^ y = x ^+ r.+1. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | modular_group_generators | |
generators_modular_groupgT (G : {group gT}) :
G \isog 'Mod_m ->
exists2 xy, extremal_generators G p n xy & modular_group_generators xy.
Proof.
case/(isoGrpP _ Grp_modular_group); rewrite card_modular_group // -/m => oG.
case/existsP=> -[x y] /= /eqP[defG xq yp xy].
rewrite norm_joinEr ?norms_cycle ?xy ?mem_cycle ... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | generators_modular_group | |
modular_group_structuregT (G : {group gT}) x y :
extremal_generators G p n (x, y) ->
G \isog 'Mod_m -> modular_group_generators (x, y) ->
let X := <[x]> in
[/\ [/\ X ><| <[y]> = G, ~~ abelian G
& {in X, forall z j, z ^ (y ^+ j) = z ^+ (j * r).+1}],
[/\ 'Z(G) = <[x ^+ p]>, 'Phi(G) = 'Z(G) & #|'... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | modular_group_structure | |
card_ext_dihedral: #|ED| = (p./2 * m)%N.
Proof. by rewrite Extremal.card // /m -mul2n -divn2 mulnA divnK. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | card_ext_dihedral | |
Grp_ext_dihedral: ED \isog Grp (x : y : x ^+ q, y ^+ p, x ^ y = x^-1).
Proof.
suffices isoED: ED \isog Grp (x : y : x ^+ q, y ^+ p, x ^ y = x ^+ q.-1).
move=> gT G; rewrite isoED.
apply: eq_existsb => [[x y]] /=; rewrite !xpair_eqE.
congr (_ && _); apply: andb_id2l; move/eqP=> xq1; congr (_ && (_ == _)).
by app... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | Grp_ext_dihedral | |
card_dihedral: #|'D_m| = m.
Proof. by rewrite /('D_m)%type def_q card_ext_dihedral ?mul1n. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | card_dihedral | |
Grp_dihedral: 'D_m \isog Grp (x : y : x ^+ q, y ^+ 2, x ^ y = x^-1).
Proof. by rewrite /('D_m)%type def_q; apply: Grp_ext_dihedral. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | Grp_dihedral | |
Grp'_dihedral: 'D_m \isog Grp (x : y : x ^+ 2, y ^+ 2, (x * y) ^+ q).
Proof.
move=> gT G; rewrite Grp_dihedral; apply/existsP/existsP=> [] [[x y]] /=.
case/eqP=> <- xq1 y2 xy; exists (x * y, y); rewrite !xpair_eqE /= eqEsubset.
rewrite !join_subG !joing_subr !cycle_subG -{3}(mulgK y x) /=.
rewrite 2?groupM ?group... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | Grp'_dihedral | |
involutions_gen_dihedralgT (x y : gT) :
let G := <<[set x; y]>> in
#[x] = 2 -> #[y] = 2 -> x != y -> G \isog 'D_#|G|.
Proof.
move=> G ox oy ne_x_y; pose q := #[x * y].
have q_gt1: q > 1 by rewrite order_gt1 -eq_invg_mul invg_expg ox.
have homG: G \homg 'D_q.*2.
rewrite Grp'_dihedral //; apply/existsP; exists (x... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | involutions_gen_dihedral | |
Grp_2dihedraln : n > 1 ->
'D_(2 ^ n) \isog Grp (x : y : x ^+ (2 ^ n.-1), y ^+ 2, x ^ y = x^-1).
Proof.
move=> n_gt1; rewrite -(ltn_predK n_gt1) expnS mul2n /=.
by apply: Grp_dihedral; rewrite (ltn_exp2l 0) // -(subnKC n_gt1).
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | Grp_2dihedral | |
card_2dihedraln : n > 1 -> #|'D_(2 ^ n)| = (2 ^ n)%N.
Proof.
move=> n_gt1; rewrite -(ltn_predK n_gt1) expnS mul2n /= card_dihedral //.
by rewrite (ltn_exp2l 0) // -(subnKC n_gt1).
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | card_2dihedral | |
card_semidihedraln : n > 3 -> #|'SD_(2 ^ n)| = (2 ^ n)%N.
Proof.
move=> n_gt3.
rewrite /('SD__)%type -(subnKC (ltnW (ltnW n_gt3))) pdiv_pfactor //.
by rewrite // !expnS !mulKn -?expnS ?Extremal.card //= (ltn_exp2l 0).
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | card_semidihedral | |
Grp_semidihedraln : n > 3 ->
'SD_(2 ^ n) \isog
Grp (x : y : x ^+ (2 ^ n.-1), y ^+ 2, x ^ y = x ^+ (2 ^ n.-2).-1).
Proof.
move=> n_gt3.
rewrite /('SD__)%type -(subnKC (ltnW (ltnW n_gt3))) pdiv_pfactor //.
rewrite !expnS !mulKn // -!expnS /=; set q := (2 ^ _)%N.
have q_gt1: q > 1 by rewrite (ltn_exp2l 0).
apply: E... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | Grp_semidihedral | |
card_quaternion: #|'Q_m| = m. Proof. by case defQ. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | card_quaternion | |
Grp_quaternion: GrpQ. Proof. by case defQ. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | Grp_quaternion | |
eq_Mod8_D8: 'Mod_8 = 'D_8. Proof. by []. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | eq_Mod8_D8 | |
generators_2dihedral:
n > 1 -> G \isog 'D_m ->
exists2 xy, extremal_generators G 2 n xy
& let: (x, y) := xy in #[y] = 2 /\ x ^ y = x^-1.
Proof.
move=> n_gt1; have [def2q _ ltqm _] := def2qr n_gt1.
case/(isoGrpP _ (Grp_2dihedral n_gt1)); rewrite card_2dihedral // -/ m => oG.
case/existsP=> -[x y] /=; re... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | generators_2dihedral | |
generators_semidihedral:
n > 3 -> G \isog 'SD_m ->
exists2 xy, extremal_generators G 2 n xy
& let: (x, y) := xy in #[y] = 2 /\ x ^ y = x ^+ r.-1.
Proof.
move=> n_gt3; have [def2q _ ltqm _] := def2qr (ltnW (ltnW n_gt3)).
case/(isoGrpP _ (Grp_semidihedral n_gt3)).
rewrite card_semidihedral // -/m => oG.
... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | generators_semidihedral | |
generators_quaternion:
n > 2 -> G \isog 'Q_m ->
exists2 xy, extremal_generators G 2 n xy
& let: (x, y) := xy in [/\ #[y] = 4, y ^+ 2 = x ^+ r & x ^ y = x^-1].
Proof.
move=> n_gt2; have [def2q def2r ltqm _] := def2qr (ltnW n_gt2).
case/(isoGrpP _ (Grp_quaternion n_gt2)); rewrite card_quaternion // -/m =... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | generators_quaternion | |
dihedral2_structure:
n > 1 -> extremal_generators G 2 n (x, y) -> G \isog 'D_m ->
[/\ [/\ X ><| Y = G, {in G :\: X, forall t, #[t] = 2}
& {in X & G :\: X, forall z t, z ^ t = z^-1}],
[/\ G ^`(1) = <[x ^+ 2]>, 'Phi(G) = G ^`(1), #|G^`(1)| = r
& nil_class G = n.-1],
'Ohm_1(G) = G /\ (for... | Theorem | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | dihedral2_structure | |
quaternion_structure:
n > 2 -> extremal_generators G 2 n (x, y) -> G \isog 'Q_m ->
[/\ [/\ pprod X Y = G, {in G :\: X, forall t, #[t] = 4}
& {in X & G :\: X, forall z t, z ^ t = z^-1}],
[/\ G ^`(1) = <[x ^+ 2]>, 'Phi(G) = G ^`(1), #|G^`(1)| = r
& nil_class G = n.-1],
[/\ 'Z(G) = <[x ^+... | Theorem | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | quaternion_structure | |
semidihedral_structure:
n > 3 -> extremal_generators G 2 n (x, y) -> G \isog 'SD_m -> #[y] = 2 ->
[/\ [/\ X ><| Y = G, #[x * y] = 4
& {in X & G :\: X, forall z t, z ^ t = z ^+ r.-1}],
[/\ G ^`(1) = <[x ^+ 2]>, 'Phi(G) = G ^`(1), #|G^`(1)| = r
& nil_class G = n.-1],
[/\ 'Z(G) = <[x ^+ r... | Theorem | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | semidihedral_structure | |
extremal_group_type:=
ModularGroup | Dihedral | SemiDihedral | Quaternion | NotExtremal. | Inductive | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | extremal_group_type | |
index_extremal_group_typec : nat :=
match c with
| ModularGroup => 0
| Dihedral => 1
| SemiDihedral => 2
| Quaternion => 3
| NotExtremal => 4
end. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | index_extremal_group_type | |
enum_extremal_groups:=
[:: ModularGroup; Dihedral; SemiDihedral; Quaternion]. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | enum_extremal_groups | |
cancel_index_extremal_groups:
cancel index_extremal_group_type (nth NotExtremal enum_extremal_groups).
Proof. by case. Qed.
Local Notation extgK := cancel_index_extremal_groups.
#[export]
HB.instance Definition _ := Countable.copy extremal_group_type (can_type extgK). | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | cancel_index_extremal_groups | |
bound_extremal_groups(c : extremal_group_type) : pickle c < 6.
Proof. by case: c. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | bound_extremal_groups | |
extremal_class(A : {set gT}) :=
let m := #|A| in let p := pdiv m in let n := logn p m in
if (n > 1) && (A \isog 'D_(2 ^ n)) then Dihedral else
if (n > 2) && (A \isog 'Q_(2 ^ n)) then Quaternion else
if (n > 3) && (A \isog 'SD_(2 ^ n)) then SemiDihedral else
if (n > 2) && (A \isog 'Mod_(p ^ n)) then ModularGro... | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | extremal_class | |
extremal2A := extremal_class A \in behead enum_extremal_groups. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | extremal2 | |
dihedral_classP:
extremal_class G = Dihedral <-> (exists2 n, n > 1 & G \isog 'D_(2 ^ n)).
Proof.
rewrite /extremal_class; split=> [ | [n n_gt1 isoG]].
by move: (logn _ _) => n; do 4?case: ifP => //; case/andP; exists n.
rewrite (card_isog isoG) card_2dihedral // -(ltn_predK n_gt1) pdiv_pfactor //.
by rewrite pfacto... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | dihedral_classP | |
quaternion_classP:
extremal_class G = Quaternion <-> (exists2 n, n > 2 & G \isog 'Q_(2 ^ n)).
Proof.
rewrite /extremal_class; split=> [ | [n n_gt2 isoG]].
by move: (logn _ _) => n; do 4?case: ifP => //; case/andP; exists n.
rewrite (card_isog isoG) card_quaternion // -(ltn_predK n_gt2) pdiv_pfactor //.
rewrite pfac... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | quaternion_classP | |
semidihedral_classP:
extremal_class G = SemiDihedral <-> (exists2 n, n > 3 & G \isog 'SD_(2 ^ n)).
Proof.
rewrite /extremal_class; split=> [ | [n n_gt3 isoG]].
by move: (logn _ _) => n; do 4?case: ifP => //; case/andP; exists n.
rewrite (card_isog isoG) card_semidihedral //.
rewrite -(ltn_predK n_gt3) pdiv_pfactor ... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | semidihedral_classP | |
odd_not_extremal2: odd #|G| -> ~~ extremal2 G.
Proof.
rewrite /extremal2 /extremal_class; case: logn => // n'.
case: andP => [[n_gt1 isoG] | _].
by rewrite (card_isog isoG) card_2dihedral ?oddX.
case: andP => [[n_gt2 isoG] | _].
by rewrite (card_isog isoG) card_quaternion ?oddX.
case: andP => [[n_gt3 isoG] | _].
... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | odd_not_extremal2 | |
modular_group_classP:
extremal_class G = ModularGroup
<-> (exists2 p, prime p &
exists2 n, n >= (p == 2) + 3 & G \isog 'Mod_(p ^ n)).
Proof.
rewrite /extremal_class; split=> [ | [p p_pr [n n_gt23 isoG]]].
move: (pdiv _) => p; set n := logn p _; do 4?case: ifP => //.
case/andP=> n_gt2 isoG _ _; rewr... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | modular_group_classP | |
extremal2_structure(gT : finGroupType) (G : {group gT}) n x y :
let cG := extremal_class G in
let m := (2 ^ n)%N in let q := (2 ^ n.-1)%N in let r := (2 ^ n.-2)%N in
let X := <[x]> in let yG := y ^: G in let xyG := (x * y) ^: G in
let My := <<yG>> in let Mxy := <<xyG>> in
extremal_generators G 2 n (x, y) -... | Theorem | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | extremal2_structure | |
maximal_cycle_extremalgT p (G X : {group gT}) :
p.-group G -> ~~ abelian G -> cyclic X -> X \subset G -> #|G : X| = p ->
(extremal_class G == ModularGroup) || (p == 2) && extremal2 G.
Proof.
move=> pG not_cGG cycX sXG iXG; rewrite /extremal2; set cG := extremal_class G.
have [|p_pr _ _] := pgroup_pdiv pG.
by ca... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | maximal_cycle_extremal | |
cyclic_SCNgT p (G U : {group gT}) :
p.-group G -> U \in 'SCN(G) -> ~~ abelian G -> cyclic U ->
[/\ p = 2, #|G : U| = 2 & extremal2 G]
\/ exists M : {group gT},
[/\ M :=: 'C_G('Mho^1(U)), #|M : U| = p, extremal_class M = ModularGroup,
'Ohm_1(M)%G \in 'E_p^2(G) & 'Ohm_1(M) \char G].
Proof.
move=> pG /SC... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | cyclic_SCN | |
normal_rank1_structuregT p (G : {group gT}) :
p.-group G -> (forall X : {group gT}, X <| G -> abelian X -> cyclic X) ->
cyclic G \/ [&& p == 2, extremal2 G & (#|G| >= 16) || (G \isog 'Q_8)].
Proof.
move=> pG dn_G_1.
have [cGG | not_cGG] := boolP (abelian G); first by left; rewrite dn_G_1.
have [X maxX]: {X | [max... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | normal_rank1_structure | |
odd_pgroup_rank1_cyclicgT p (G : {group gT}) :
p.-group G -> odd #|G| -> cyclic G = ('r_p(G) <= 1).
Proof.
move=> pG oddG; rewrite -rank_pgroup //; apply/idP/idP=> [cycG | dimG1].
by rewrite -abelian_rank1_cyclic ?cyclic_abelian.
have [X nsXG cXX|//|] := normal_rank1_structure pG; last first.
by rewrite (negPf (o... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | odd_pgroup_rank1_cyclic | |
prime_Ohm1PgT p (G : {group gT}) :
p.-group G -> G :!=: 1 ->
reflect (#|'Ohm_1(G)| = p)
(cyclic G || (p == 2) && (extremal_class G == Quaternion)).
Proof.
move=> pG ntG; have [p_pr p_dvd_G _] := pgroup_pdiv pG ntG.
apply: (iffP idP) => [|oG1p].
case/orP=> [cycG|]; first exact: Ohm1_cyclic_pgroup_prime... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | prime_Ohm1P | |
symplectic_type_group_structuregT p (G : {group gT}) :
p.-group G -> (forall X : {group gT}, X \char G -> abelian X -> cyclic X) ->
exists2 E : {group gT}, E :=: 1 \/ extraspecial E
& exists R : {group gT},
[/\ cyclic R \/ [/\ p = 2, extremal2 R & #|R| >= 16],
E \* R = G
& E :&: R = 'Z(E)].
Pr... | Theorem | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset prime binomial",
"From mathcomp Require Import fingroup morphism perm automorphism presentation",
"From mathcomp Require Import quotie... | solvable/extremal.v | symplectic_type_group_structure | |
fmod_of(gT : finGroupType) (A : {group gT}) (abelA : abelian A) :=
Fmod x & x \in A.
Bind Scope ring_scope with fmod_of. | Inductive | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | fmod_of | |
fmvalu := val (f2sub_magma u).
#[export]
HB.instance Definition _ := [isSub for fmval].
Local Notation valA := (val: fmodA -> gT) (only parsing).
#[export]
HB.instance Definition _ := [Finite of fmodA by <:]. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | fmval | |
fmodx := sub2f (subg A x). | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | fmod | |
actru x := if x \in 'N(A) then fmod (fmval u ^ x) else u. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | actr | |
fmod_oppu := sub2f u^-1. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | fmod_opp | |
fmod_addu v := sub2f (u * v).
Fact fmod_add0r : left_id (sub2f 1) fmod_add.
Proof. by move=> u; apply: val_inj; apply: mul1g. Qed.
Fact fmod_addrA : associative fmod_add.
Proof. by move=> u v w; apply: val_inj; apply: mulgA. Qed.
Fact fmod_addNr : left_inverse (sub2f 1) fmod_opp fmod_add.
Proof. by move=> u; apply: val... | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | fmod_add | |
Definition_ := [finGroupMixin of fmodA for +%R]. | HB.instance | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | Definition | |
fmodPu : val u \in A. Proof. exact: valP. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | fmodP | |
fmod_inj: injective fmval. Proof. exact: val_inj. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | fmod_inj | |
congr_fmodu v : u = v -> fmval u = fmval v.
Proof. exact: congr1. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | congr_fmod | |
fmvalA: {morph valA : x y / x + y >-> (x * y)%g}. Proof. by []. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | fmvalA | |
fmvalN: {morph valA : x / - x >-> x^-1%g}. Proof. by []. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | fmvalN | |
fmval0: valA 0 = 1%g. Proof. by []. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | fmval0 | |
fmval_morphism:= @Morphism _ _ setT fmval (in2W fmvalA). | Canonical | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | fmval_morphism | |
fmval_sum:= big_morph fmval fmvalA fmval0. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | fmval_sum | |
fmvalZn : {morph valA : x / x *+ n >-> (x ^+ n)%g}.
Proof. by move=> u; rewrite /= morphX ?inE. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | fmvalZ | |
fmodKcondx : val (fmod x) = if x \in A then x else 1%g.
Proof. by rewrite /= /fmval /= val_insubd. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | fmodKcond | |
fmodK: {in A, cancel fmod val}. Proof. exact: subgK. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | fmodK | |
fmvalK: cancel val fmod.
Proof. by case=> x Ax; apply: val_inj; rewrite /fmod /= sgvalK. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | fmvalK | |
fmod1: fmod 1 = 0. Proof. by rewrite -fmval0 fmvalK. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | fmod1 | |
fmodM: {in A &, {morph fmod : x y / (x * y)%g >-> x + y}}.
Proof. by move=> x y Ax Ay /=; apply: val_inj; rewrite /fmod morphM. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | fmodM | |
fmod_morphism:= Morphism fmodM. | Canonical | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | fmod_morphism | |
fmodXn : {in A, {morph fmod : x / (x ^+ n)%g >-> x *+ n}}.
Proof. exact: morphX. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | fmodX | |
fmodV: {morph fmod : x / x^-1%g >-> - x}.
Proof.
move=> x; apply: val_inj; rewrite fmvalN !fmodKcond groupV.
by case: (x \in A); rewrite ?invg1.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | fmodV | |
injm_fmod: 'injm fmod.
Proof.
by apply/injmP=> x y Ax Ay []; move/val_inj; apply: (injmP (injm_subg A)).
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | injm_fmod | |
fmvalJcondu x :
val (u ^@ x) = if x \in 'N(A) then val u ^ x else val u.
Proof. by case: ifP => Nx; rewrite /actr Nx ?fmodK // memJ_norm ?fmodP. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | fmvalJcond | |
fmvalJu x : x \in 'N(A) -> val (u ^@ x) = val u ^ x.
Proof. by move=> Nx; rewrite fmvalJcond Nx. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | fmvalJ | |
fmodJx y : y \in 'N(A) -> fmod (x ^ y) = fmod x ^@ y.
Proof.
move=> Ny; apply: val_inj; rewrite fmvalJ ?fmodKcond ?memJ_norm //.
by case: ifP => // _; rewrite conj1g.
Qed.
Fact actr_is_action : is_action 'N(A) actr.
Proof.
split=> [a u v eq_uv_a | u a b Na Nb].
case Na: (a \in 'N(A)); last by rewrite /actr Na in eq_u... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | fmodJ | |
actr_action:= Action actr_is_action. | Canonical | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | actr_action | |
act0rx : 0 ^@ x = 0.
Proof. by rewrite /actr conj1g morph1 if_same. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | act0r | |
actArx : {morph actr^~ x : u v / u + v}.
Proof.
by move=> u v; apply: val_inj; rewrite !(fmvalA, fmvalJcond) conjMg; case: ifP.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | actAr | |
actr_sumx := big_morph _ (actAr x) (act0r x). | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | actr_sum | |
actNrx : {morph actr^~ x : u / - u}.
Proof. by move=> u; apply: (addrI (u ^@ x)); rewrite -actAr !subrr act0r. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | actNr | |
actZrx n : {morph actr^~ x : u / u *+ n}.
Proof.
by move=> u; elim: n => [|n IHn]; rewrite ?act0r // !mulrS actAr IHn.
Qed.
Fact actr_is_groupAction : is_groupAction setT 'M.
Proof.
move=> a Na /[1!inE]; apply/andP; split; first by apply/subsetP=> u _ /[1!inE].
by apply/morphicP=> u v _ _; rewrite !permE /= actAr.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | actZr | |
actr_groupAction:= GroupAction actr_is_groupAction. | Canonical | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | actr_groupAction | |
actr1u : u ^@ 1 = u.
Proof. exact: act1. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | actr1 | |
actrM: {in 'N(A) &, forall x y u, u ^@ (x * y) = u ^@ x ^@ y}.
Proof.
by move=> x y Nx Ny /= u; apply: val_inj; rewrite !fmvalJ ?conjgM ?groupM.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | actrM | |
actrKx : cancel (actr^~ x) (actr^~ x^-1%g).
Proof.
move=> u; apply: val_inj; rewrite !fmvalJcond groupV.
by case: ifP => -> //; rewrite conjgK.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | actrK | |
actrKVx : cancel (actr^~ x^-1%g) (actr^~ x).
Proof. by move=> u; rewrite /= -{2}(invgK x) actrK. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | actrKV | |
Gaschutz_split: [splits G, over H] = [splits P, over H].
Proof.
apply/splitsP/splitsP=> [[K /complP[tiHK eqHK]] | [Q /complP[tiHQ eqHQ]]].
exists (K :&: P)%G; rewrite inE setICA (setIidPl sHP) setIC tiHK eqxx.
by rewrite group_modl // eqHK (sameP eqP setIidPr).
have sQP: Q \subset P by rewrite -eqHQ mulG_subr.
pose... | Theorem | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | Gaschutz_split | |
Gaschutz_transitive: {in [complements to H in G] &,
forall K L, K :&: P = L :&: P -> exists2 x, x \in H & L :=: K :^ x}.
Proof.
move=> K L /=; set Q := K :&: P => /complP[tiHK eqHK] cpHL QeqLP.
have [trHL eqHL] := complP cpHL.
pose nu x := fmod (divgr H L x^-1).
have sKG: {subset K <= G} by apply/subsetP; rewrite -e... | Theorem | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | Gaschutz_transitive | |
coprime_abel_cent_TI(gT : finGroupType) (A G : {group gT}) :
A \subset 'N(G) -> coprime #|G| #|A| -> abelian G -> 'C_[~: G, A](A) = 1.
Proof.
move=> nGA coGA abG; pose f x := val (\sum_(a in A) fmod abG x ^@ a)%R.
have fM: {in G &, {morph f : x y / x * y}}.
move=> x y Gx Gy /=; rewrite -fmvalA -big_split /=; congr ... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | coprime_abel_cent_TI | |
transferg := V repr g. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | transfer | |
transferM: {in G &, {morph transfer : x y / (x * y)%g >-> x + y}}.
Proof.
move=> s t Gs Gt /=.
rewrite [transfer t](reindex_acts 'Rs _ Gs) ?actsRs_rcosets //= -big_split /=.
apply: eq_bigr => _ /rcosetsP[x Gx ->]; rewrite !rcosetE -!rcosetM.
rewrite -zmodMgE -morphM -?mem_rcoset; first by rewrite !mulgA mulgKV rcosetM.... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | transferM | |
transfer_morphism:= Morphism transferM. | Canonical | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | transfer_morphism | |
transfer_indepX (rX := transversal_repr 1 X) :
is_transversal X HG G -> {in G, transfer =1 V rX}.
Proof.
move=> trX g Gg; have mem_rX := repr_mem_pblock trX 1; rewrite -/rX in mem_rX.
apply: (addrI (\sum_(Hx in HG) fmalpha (repr Hx * (rX Hx)^-1))).
rewrite {1}(reindex_acts 'Rs _ Gg) ?actsRs_rcosets // -!big_split /=.... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | transfer_indep | |
mulg_exp_card_rcosetsx : x * (g ^+ n_ x) \in H :* x.
Proof.
rewrite /n_ /indexg -orbitRs -porbit_actperm ?inE //.
rewrite -{2}(iter_porbit (actperm 'Rs g) (H :* x)) -permX -morphX ?inE //.
by rewrite actpermE //= rcosetE -rcosetM rcoset_refl.
Qed.
Let HGg : {set {set {set gT}}} := orbit 'Rs <[g]> @: HG.
Let partHG : pa... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | mulg_exp_card_rcosets | |
rcosets_cycle_partition: partition (HG :* <[g]>) G.
Proof. by rewrite defHGg; have [] := partition_partition partHG partHGg. Qed.
Variable X : {set gT}.
Hypothesis trX : is_transversal X (HG :* <[g]>) G.
Let sXG : {subset X <= G}. Proof. exact/subsetP/(transversal_sub trX). Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | rcosets_cycle_partition | |
rcosets_cycle_transversal: H_g_rcosets @: X = HGg.
Proof.
have sHXgHGg x: x \in X -> H_g_rcosets x \in HGg.
by move/sXG=> Gx; apply: imset_f; rewrite -rcosetE imset_f.
apply/setP=> Hxg; apply/imsetP/idP=> [[x /sHXgHGg HGgHxg -> //] | HGgHxg].
have [_ /rcosetsP[z Gz ->] ->] := imsetP HGgHxg.
pose Hzg := H :* z * <[g]>... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | rcosets_cycle_transversal | |
sum_index_rcosets_cycle: (\sum_(x in X) n_ x)%N = #|G : H|.
Proof. by rewrite [#|G : H|](card_partition partHGg) -defHgX big_imset. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | sum_index_rcosets_cycle | |
transfer_cycle_expansion:
transfer g = \sum_(x in X) fmalpha ((g ^+ n_ x) ^ (x^-1)).
Proof.
pose Y := \bigcup_(x in X) [set x * g ^+ i | i : 'I_(n_ x)].
pose rY := transversal_repr 1 Y.
pose pcyc x := porbit (actperm 'Rs g) (H :* x).
pose traj x := traject (actperm 'Rs g) (H :* x) #|pcyc x|.
have Hgr_eq x: H_g_rcose... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype bigop ssralg finset fingroup",
"From mathcomp Require Import morphism perm finalg action gproduct commutator ",
"From mathcomp Require Import ... | solvable/finmodule.v | transfer_cycle_expansion | |
semiregularK H := {in H^#, forall x, 'C_K[x] = 1}. | Definition | solvable | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] | solvable/frobenius.v | semiregular | |
semiprimeK H := {in H^#, forall x, 'C_K[x] = 'C_K(H)}. | Definition | solvable | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] | solvable/frobenius.v | semiprime | |
normedTIA G L := [&& A != set0, trivIset (A :^: G) & 'N_G(A) == L]. | Definition | solvable | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] | solvable/frobenius.v | normedTI | |
Frobenius_group_with_complementG H := (H != G) && normedTI H^# G H. | Definition | solvable | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] | solvable/frobenius.v | Frobenius_group_with_complement | |
Frobenius_groupG :=
[exists H : {group gT}, Frobenius_group_with_complement G H]. | Definition | solvable | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] | solvable/frobenius.v | Frobenius_group | |
Frobenius_group_with_kernel_and_complementG K H :=
(K ><| H == G) && Frobenius_group_with_complement G H. | Definition | solvable | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] | solvable/frobenius.v | Frobenius_group_with_kernel_and_complement | |
Frobenius_group_with_kernelG K :=
[exists H : {group gT}, Frobenius_group_with_kernel_and_complement G K H]. | Definition | solvable | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div",
"From mathcomp Require Import fintype bigop prime finset fingroup morphism",
"From mathcomp Require Import perm action quotient gproduct cyclic center",
"From mathcomp Require Import pgroup nilpotent sylow hall abelian"
] | solvable/frobenius.v | Frobenius_group_with_kernel |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.