statement stringlengths 1 4.33k | proof stringlengths 0 37.9k | type stringclasses 25
values | symbolic_name stringlengths 1 67 | library stringclasses 10
values | filename stringclasses 112
values | imports listlengths 2 138 | deps listlengths 0 64 | docstring stringclasses 798
values | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
rank_DnQ n : 'r_2('D^n*Q) = n.+1. | Proof.
have pDnQ: 2.-group 'D^n*Q := DnQ_pgroup n.
have esDnQ: extraspecial 'D^n*Q := DnQ_extraspecial n.
do [case: DnQ_P => gz isoZ; set DnQ := setT] in pDnQ esDnQ *.
suffices [E]: exists2 E, E \in 'E*_2(DnQ) & logn 2 #|E| = n.+1.
by rewrite (pmaxElem_extraspecial pDnQ esDnQ); case/pnElemP=> _ _ <-.
have oZ: #|'Z(Dn... | Lemma | rank_DnQ | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"DnQ_P",
"DnQ_extraspecial",
"DnQ_pgroup",
"Lagrange",
"Ohm1_id",
"OhmE",
"abelE",
"abelem",
"abelem_pgroup",
"abelianE",
"apply",
"cardG_gt0",
"card_center_extraspecial",
"card_injm",
"card_pX1p2n",
"card_subcent_extraspecial",
"centS",
"center_ncprod0",
"cpair1g",
"cpair1g_ce... | The second concluding remark of Aschbacher (23.14). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
not_isog_Dn_DnQ n : ~~ ('D^n \isog 'D^n.-1*Q). | Proof.
case: n => [|n] /=; first by rewrite isogEcard card_pX1p2n // card_DnQ andbF.
apply: contraL (leqnn n.+1) => isoDn1DnQ.
by rewrite -ltnNge -rank_Dn (isog_p_rank isoDn1DnQ) rank_DnQ.
Qed. | Lemma | not_isog_Dn_DnQ | solvable | solvable/extraspecial.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gproduct",
"gfunctor",
"ss... | [
"apply",
"card_DnQ",
"card_pX1p2n",
"isog",
"isogEcard",
"isog_p_rank",
"leqnn",
"ltnNge",
"rank_Dn",
"rank_DnQ"
] | The final concluding remark of Aschbacher (23.14). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
a : 'Z_p | := Zp1. | Let | a | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Zp1"
] | #[s] %| p and s 1%R = 1%R ^+ e. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
b : 'Z_q | := Zp1. | Let | b | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Zp1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
B | := <[b]>. | Notation | B | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
aut_of | :=
odflt 1 [pick s in Aut B | p > 1 & (#[s] %| p) && (s b == b ^+ e)]. | Definition | aut_of | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Aut",
"pick"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
aut_dvdn : #[aut_of] %| #[a]. | Proof.
rewrite order_Zp1 /aut_of; case: pickP => [s | _]; last by rewrite order1.
by case/and4P=> _ p_gt1 p_s _; rewrite Zp_cast.
Qed. | Lemma | aut_dvdn | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Zp_cast",
"aut_of",
"last",
"order1",
"order_Zp1",
"p_gt1",
"pickP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
act_morphism | := eltm_morphism aut_dvdn. | Definition | act_morphism | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"aut_dvdn",
"eltm_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
base_act | := ([Aut B] \o act_morphism)%gact. | Definition | base_act | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Aut",
"act_morphism",
"gact"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
act_dom : <[a]> \subset act_dom base_act. | Proof.
rewrite cycle_subG 2!inE cycle_id /= eltm_id /aut_of.
by case: pickP => [op /andP[] | _] //=; rewrite group1.
Qed. | Lemma | act_dom | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"aut_of",
"base_act",
"cycle_id",
"cycle_subG",
"eltm_id",
"group1",
"inE",
"pickP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gact | := (base_act \ act_dom)%gact. | Definition | gact | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"act_dom",
"base_act"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gtype_unlockable | := Unlockable gtype.unlock. | Canonical | gtype_unlockable | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"gtype"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gtype | := (gtype q p e) (only parsing). | Notation | gtype | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gact | := (gact q p e) (only parsing). | Notation | gact | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
aut_of | := (aut_of q p e) (only parsing). | Notation | aut_of | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
(p_gt1 : p > 1) (q_gt1 : q > 1). | Hypotheses | p_gt1 | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"q_gt1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
card : #|[set: gtype]| = (p * q)%N. | Proof.
rewrite [gtype.body]unlock -(sdprod_card (sdprod_sdpair _)).
rewrite !card_injm ?injm_sdpair1 ?injm_sdpair2 //.
by rewrite mulnC -!orderE !order_Zp1 !Zp_cast.
Qed. | Lemma | card | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Zp_cast",
"body",
"card_injm",
"gtype",
"injm_sdpair1",
"injm_sdpair2",
"mulnC",
"orderE",
"order_Zp1",
"sdprod_card",
"sdprod_sdpair"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Grp : (exists s, [/\ s \in Aut B, #[s] %| p & s b = b ^+ e]) ->
[set: gtype] \isog Grp (x : y : x ^+ q, y ^+ p, x ^ y = x ^+ e). | Proof.
rewrite [gtype.body]unlock => [[s [AutBs dvd_s_p sb]]].
have memB: _ \in B by move=> c; rewrite -Zp_cycle inE.
have Aa: a \in <[a]> by rewrite !cycle_id.
have [oa ob]: #[a] = p /\ #[b] = q by rewrite !order_Zp1 !Zp_cast.
have def_s: aut_of = s.
rewrite /aut_of; case: pickP => /= [t | ]; last first.
by move... | Lemma | Grp | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Aut",
"Zp_cast",
"Zp_cycle",
"act",
"apermE",
"apply",
"aut_of",
"autact",
"autmE",
"body",
"conjXg",
"conjg1",
"conjgM",
"cycleP",
"cycle_id",
"defG",
"dvdnn",
"eltm",
"eltmE",
"eltm_id",
"eq_Aut",
"eqxx",
"existsP",
"expgS",
"fP",
"gT",
"gact",
"gactX",
"gr... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
p | := pdiv m. | Let | p | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"pdiv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
q | := m %/ p. | Let | q | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modular_gtype | := gtype q p (q %/ p).+1. | Definition | modular_gtype | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"gtype"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dihedral_gtype | := gtype q 2 q.-1. | Definition | dihedral_gtype | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"gtype"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
semidihedral_gtype | := gtype q 2 (q %/ p).-1. | Definition | semidihedral_gtype | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"gtype"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quaternion_kernel | :=
<<[set u | u ^+ 2 == 1] :\: [set u ^+ 2 | u in [set: gtype q 4 q.-1]]>>. | Definition | quaternion_kernel | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"gtype"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quaternion_unlock | := Unlockable quaternion_gtype.unlock. | Canonical | quaternion_unlock | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''Mod_' m" | := (modular_gtype m) : type_scope. | Notation | ''Mod_' m | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"modular_gtype"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''Mod_' m" | := [set: gsort 'Mod_m] : group_scope. | Notation | ''Mod_' m | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"gsort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''Mod_' m" | := [set: gsort 'Mod_m]%G : Group_scope. | Notation | ''Mod_' m | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"gsort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''D_' m" | := (dihedral_gtype m) : type_scope. | Notation | ''D_' m | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"dihedral_gtype"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''D_' m" | := [set: gsort 'D_m] : group_scope. | Notation | ''D_' m | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"gsort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''D_' m" | := [set: gsort 'D_m]%G : Group_scope. | Notation | ''D_' m | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"gsort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''SD_' m" | := (semidihedral_gtype m) : type_scope. | Notation | ''SD_' m | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"semidihedral_gtype"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''SD_' m" | := [set: gsort 'SD_m] : group_scope. | Notation | ''SD_' m | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"gsort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''SD_' m" | := [set: gsort 'SD_m]%G : Group_scope. | Notation | ''SD_' m | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"gsort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''Q_' m" | := (quaternion_gtype m) : type_scope. | Notation | ''Q_' m | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''Q_' m" | := [set: gsort 'Q_m] : group_scope. | Notation | ''Q_' m | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"gsort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''Q_' m" | := [set: gsort 'Q_m]%G : Group_scope. | Notation | ''Q_' m | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"gsort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cyclic_pgroup_Aut_structure gT p (G : {group gT}) :
p.-group G -> cyclic G -> G :!=: 1 ->
let q := #|G| in let n := (logn p q).-1 in
let A := Aut G in let P := 'O_p(A) in let F := 'O_p^'(A) in
exists m : {perm gT} -> 'Z_q,
[/\ [/\ {in A & G, forall a x, x ^+ m a = a x},
m 1 = 1%R /\ {in A &, {morp... | Proof.
move=> pG cycG ntG q n0 A P F; have [p_pr p_dvd_G [n oG]] := pgroup_pdiv pG ntG.
have [x0 defG] := cyclicP cycG; have Gx0: x0 \in G by rewrite defG cycle_id.
rewrite {1}/q oG pfactorK //= in n0 *; rewrite {}/n0.
have [p_gt1 min_p] := primeP p_pr; have p_gt0 := ltnW p_gt1.
have q_gt1: q > 1 by rewrite cardG_gt1.
... | Lemma | cyclic_pgroup_Aut_structure | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Aut",
"Aut_cyclic_abelian",
"Aut_prime_cyclic",
"Fp_nat_mod",
"Hall",
"Lagrange",
"Ohm1_cyclic_pgroup_prime",
"OhmE",
"Ohm_sub",
"Sub",
"Sylow",
"TI_cardMg",
"Uu",
"Zp_cast",
"Zp_unit_isog",
"Zp_unit_isom",
"Zp_unit_morphism",
"abelian",
"abelian_nil",
"add1n",
"addSn",
"a... | the inverting involution is available for all non-trivial p-cycles. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
extremal_generators gT (A : {set gT}) p n xy | :=
let: (x, y) := xy in
[/\ #|A| = (p ^ n)%N, x \in A, #[x] = (p ^ n.-1)%N & y \in A :\: <[x]>]. | Definition | extremal_generators | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"gT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
extremal_generators_facts gT (G : {group gT}) p n x y :
prime p -> extremal_generators G p n (x, y) ->
[/\ p.-group G, maximal <[x]> G, <[x]> <| G,
<[x]> * <[y]> = G & <[y]> \subset 'N(<[x]>)]. | Proof.
move=> p_pr [oG Gx ox] /setDP[Gy notXy].
have pG: p.-group G by rewrite /pgroup oG pnatX pnat_id.
have maxX: maximal <[x]> G.
rewrite p_index_maximal -?divgS ?cycle_subG // -orderE oG ox.
case: (n) oG => [|n' _]; last by rewrite -expnB ?subSnn ?leqnSn ?prime_gt0.
move/eqP; rewrite -trivg_card1; case/trivgP... | Lemma | extremal_generators_facts | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"apply",
"cycle_subG",
"divgS",
"expnB",
"extremal_generators",
"gT",
"group",
"group1_contra",
"last",
"leqnSn",
"maximal",
"mulg_normal_maximal",
"n'",
"normal_norm",
"orderE",
"pG",
"p_index_maximal",
"p_maximal_normal",
"p_pr",
"pgroup",
"pnatX",
"pnat_id",
"prime",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
m | := (p ^ n)%N. | Let | m | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
q | := (p ^ n.-1)%N. | Let | q | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r | := (p ^ n.-2)%N. | Let | r | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
(p_pr : prime p) (n_gt2 : n > 2). | Hypotheses | p_pr | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"n_gt2",
"prime"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
def_n | := esym (subnKC n_gt2). | Let | def_n | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"n_gt2",
"subnKC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
def_p : pdiv m = p. | Proof. by rewrite /m def_n pdiv_pfactor. Qed. | Let | def_p | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"def_n",
"pdiv",
"pdiv_pfactor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
def_q : m %/ p = q. | Proof. by rewrite /m /q def_n expnS mulKn. Qed. | Let | def_q | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"def_n",
"expnS",
"mulKn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
def_r : q %/ p = r. | Proof. by rewrite /r /q def_n expnS mulKn. Qed. | Let | def_r | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"def_n",
"expnS",
"mulKn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltqm : q < m. | Proof. by rewrite ltn_exp2l // def_n. Qed. | Let | ltqm | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"def_n",
"ltn_exp2l"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltrq : r < q. | Proof. by rewrite ltn_exp2l // def_n. Qed. | Let | ltrq | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"def_n",
"ltn_exp2l"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r_gt0 : 0 < r. | Proof. by rewrite expn_gt0 ?p_gt0. Qed. | Let | r_gt0 | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"expn_gt0",
"p_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
q_gt1 : q > 1. | Proof. exact: leq_ltn_trans r_gt0 ltrq. Qed. | Let | q_gt1 | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"leq_ltn_trans",
"ltrq",
"r_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_modular_group : #|'Mod_(p ^ n)| = (p ^ n)%N. | Proof. by rewrite Extremal.card def_p ?def_q // -expnS def_n. Qed. | Lemma | card_modular_group | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"card",
"def_n",
"def_p",
"def_q",
"expnS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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.
have pB: p.-group B by rewrite /pgroup oB pnatX ?pnat_id.
have ntB: B != 1 by rewrite -c... | Lemma | Grp_modular_group | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Grp",
"Zp1",
"Zp_cast",
"addn2",
"apply",
"cardG_gt1",
"cycle_cyclic",
"cycle_id",
"cyclic",
"cyclic_pgroup_Aut_structure",
"def_n",
"def_p",
"def_q",
"def_r",
"eqSS",
"even_prime",
"expg_znat",
"group",
"isog",
"last",
"ltnW",
"modular_gtype",
"orderE",
"order_Zp1",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modular_group_generators gT (xy : gT * gT) | :=
let: (x, y) := xy in #[y] = p /\ x ^ y = x ^+ r.+1. | Definition | modular_group_generators | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"gT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
generators_modular_group gT (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 // in defG.
have [Gx Gy]: x \in G /\ y \in G.
by apply/andP; rewrite -!cycle_subG -mulG_subG defG.
have notXy: y \notin <[x]>.
apply: cont... | Lemma | generators_modular_group | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Grp_modular_group",
"TI_cardMg",
"apply",
"card_modular_group",
"cycle_subG",
"defG",
"dvdn_leq",
"eqn_pmul2r",
"existsP",
"expnSr",
"extremal_generators",
"gT",
"group",
"group1_contra",
"inE",
"isoGrpP",
"isog",
"leqNgt",
"ltnW",
"ltn_predK",
"ltqm",
"mem_cycle",
"modu... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modular_group_structure gT (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) & #|... | Proof.
move=> genG isoG [oy xy] X.
have [oG Gx ox /setDP[Gy notXy]] := genG; rewrite -/m -/q in ox oG.
have [pG _ nsXG defXY nXY] := extremal_generators_facts p_pr genG.
have [sXG nXG] := andP nsXG; have sYG: <[y]> \subset G by rewrite cycle_subG.
have n1_gt1: n.-1 > 1 by [rewrite def_n]; have n1_gt0 := ltnW n1_gt1.
ha... | Lemma | modular_group_structure | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Lagrange",
"Mho",
"MhoE",
"Mho_p_elt",
"OhmE",
"OhmS",
"Ohm_dprod",
"Ohm_id",
"Ohm_leq",
"Ohm_p_cycle",
"Ohm_sub",
"Phi_joing",
"TI_cardMg",
"abelian",
"abelianM",
"actX",
"add1n",
"addSn",
"addnA",
"apply",
"bin2odd",
"cardG_gt0",
"card_quotient",
"cent1C",
"cent1P"... | - We corrected a pair of typos. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
q_gt1 : q > 1. | Hypothesis | q_gt1 | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
m | := q.*2. | Let | m | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
def2 : pdiv m = 2. | Proof.
apply/eqP; rewrite /m -mul2n eqn_leq pdiv_min_dvd ?dvdn_mulr //.
by rewrite prime_gt1 // pdiv_prime // (@leq_pmul2l 2 1) ltnW.
Qed. | Let | def2 | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"apply",
"dvdn_mulr",
"eqn_leq",
"leq_pmul2l",
"ltnW",
"mul2n",
"pdiv",
"pdiv_min_dvd",
"pdiv_prime",
"prime_gt1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
def_q : m %/ pdiv m = q. | Proof. by rewrite def2 divn2 half_double. Qed. | Let | def_q | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"def2",
"divn2",
"half_double",
"pdiv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
(p_gt1 : p > 1) (even_p : 2 %| p). | Hypotheses | p_gt1 | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
ED | := [set: gsort (Extremal.gtype q p q.-1)]. | Notation | ED | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"gsort",
"gtype"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_ext_dihedral : #|ED| = (p./2 * m)%N. | Proof. by rewrite Extremal.card // /m -mul2n -divn2 mulnA divnK. Qed. | Lemma | card_ext_dihedral | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"ED",
"card",
"divn2",
"divnK",
"mul2n",
"mulnA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 apply/eqP; rewrite eq_sym eq_invg_mul -expgS (ltn_predK q_gt1) xq1.
have u... | Lemma | Grp_ext_dihedral | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Aut_aut",
"ED",
"Grp",
"Zp1",
"Zp_cast",
"Zp_unitm",
"apply",
"autE",
"cycle_id",
"dvdn_trans",
"eq_existsb",
"eq_expg_mod_order",
"eq_invg_mul",
"eq_sym",
"expgS",
"gT",
"in_setT",
"injm_Zp_unitm",
"isog",
"ltn_predK",
"modn_mod",
"modn_small",
"mulrNN",
"order_Zp1",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_dihedral : #|'D_m| = m. | Proof. by rewrite /('D_m)%type def_q card_ext_dihedral ?mul1n. Qed. | Lemma | card_dihedral | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"card_ext_dihedral",
"def_q",
"mul1n",
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | Grp_dihedral | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Grp",
"Grp_ext_dihedral",
"apply",
"def_q",
"isog",
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 ?groupV ?mem_gen ?inE ?cycle_id ?orbT //= -mulgA expgS.
by rewrite {1}(con... | Lemma | Grp'_dihedral | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Grp",
"Grp_dihedral",
"apply",
"conjgC",
"cycle_id",
"cycle_subG",
"eqEsubset",
"eq_invg_mul",
"eq_sym",
"eqxx",
"existsP",
"expgS",
"gT",
"groupM",
"groupV",
"inE",
"isog",
"join_subG",
"joing_subr",
"mem_gen",
"mulKg",
"mulg1",
"mulgA",
"mulgK",
"xpair_eqE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
involutions_gen_dihedral gT (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, y); rewrite /= !xpair_eqE.
by rewrite joing_idl joing_idr -{1}ox -oy !expg_order !eqxx.
suff oG: #|G| = q.*2 by rewrite ... | Lemma | involutions_gen_dihedral | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Grp'_dihedral",
"apply",
"cardSg",
"card_dihedral",
"card_homg",
"cycle2g",
"cycle_subG",
"cyclic",
"cyclicP",
"dvdnP",
"dvdn_divisors",
"dvdn_pmul2r",
"eqEcard",
"eq_invg_mul",
"eq_subG_cyclic",
"eq_sym",
"eqxx",
"existsP",
"expg_order",
"gT",
"genS",
"groupM",
"homg",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Grp_2dihedral n : 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 | Grp_2dihedral | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Grp",
"Grp_dihedral",
"apply",
"expnS",
"isog",
"ltn_exp2l",
"ltn_predK",
"mul2n",
"subnKC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_2dihedral n : 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 | card_2dihedral | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"card_dihedral",
"expnS",
"ltn_exp2l",
"ltn_predK",
"mul2n",
"subnKC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_semidihedral n : 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 | card_semidihedral | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"card",
"expnS",
"ltnW",
"ltn_exp2l",
"mulKn",
"pdiv_pfactor",
"subnKC",
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Grp_semidihedral n : 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: Extremal.Grp => //; set B := <[_]>.
have oB: #|B| = q by rewrite -orderE order_Zp1 Zp_cast.
have pB: 2.-group B by rewrite... | Lemma | Grp_semidihedral | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Aut",
"Grp",
"Zp_cast",
"Zp_expg",
"Zp_nat",
"apply",
"cardG_gt1",
"cent1id",
"centM",
"centP",
"cent_cycle",
"centsC",
"commute",
"cycle_cyclic",
"cycle_id",
"cyclic_pgroup_Aut_structure",
"dprodP",
"expgMn",
"expg_order",
"expnS",
"group",
"groupM",
"inE",
"isog",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
n_gt2 : n > 2. | Hypothesis | n_gt2 | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
m | := (2 ^ n)%N. | Let | m | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
q | := (2 ^ n.-1)%N. | Let | q | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r | := (2 ^ n.-2)%N. | Let | r | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
GrpQ | := 'Q_m \isog Grp (x : y : x ^+ q, y ^+ 2 = x ^+ r, x ^ y = x^-1). | Let | GrpQ | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Grp",
"isog"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
defQ : #|'Q_m| = m /\ GrpQ. | Proof.
have q_gt1 : q > 1 by rewrite (ltn_exp2l 0) // -(subnKC n_gt2).
have def_m : (2 * q)%N = m by rewrite -expnS (ltn_predK n_gt2).
have def_q : m %/ pdiv m = q
by rewrite /m -(ltn_predK n_gt2) pdiv_pfactor // expnS mulKn.
have r_gt1 : r > 1 by rewrite (ltn_exp2l 0) // -(subnKC n_gt2).
have def2r : (2 * r)%N = q b... | Let | defQ | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Grp",
"GrpQ",
"Grp_ext_dihedral",
"TI_cardMg",
"add0n",
"addnC",
"addnn",
"apply",
"canF_eq",
"cardG_gt0",
"cardMg_TI",
"card_ext_dihedral",
"card_quotient",
"commuteX",
"conjVg",
"conjXg",
"conjg1",
"conjgC",
"conjgE",
"conjgK",
"conjgM",
"conjg_set1",
"coset",
"coset... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_quaternion : #|'Q_m| = m. | Proof. by case defQ. Qed. | Lemma | card_quaternion | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"defQ"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Grp_quaternion : GrpQ. | Proof. by case defQ. Qed. | Lemma | Grp_quaternion | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"GrpQ",
"defQ"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_Mod8_D8 : 'Mod_8 = 'D_8. | Proof. by []. Qed. | Lemma | eq_Mod8_D8 | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
q_gt0: q > 0. | Proof. by rewrite expn_gt0. Qed. | Let | q_gt0 | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"expn_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r_gt0: r > 0. | Proof. by rewrite expn_gt0. Qed. | Let | r_gt0 | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"expn_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
def2qr : n > 1 -> [/\ 2 * q = m, 2 * r = q, q < m & r < q]%N. | Proof. by rewrite /q /m /r; move/subnKC=> <-; rewrite !ltn_exp2l ?expnS. Qed. | Let | def2qr | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"expnS",
"ltn_exp2l",
"subnKC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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] /=; rewrite -/q => /eqP[defG xq y2 xy].
have{} defG: <[x]> * <[y]> = G.
by rewrite -norm_joinEr // norms_cycle xy groupV cycle_id.
have notXy: y \notin <[x]>.... | Lemma | generators_2dihedral | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Grp_2dihedral",
"TI_cardMg",
"apply",
"card_2dihedral",
"cycle_id",
"cycle_subG",
"def2qr",
"defG",
"double_inj",
"dvdn_leq",
"existsP",
"extremal_generators",
"group1_contra",
"groupV",
"inE",
"isoGrpP",
"isog",
"leqNgt",
"ltqm",
"mul2n",
"mulGSid",
"mulG_subl",
"mulG_s... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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.
case/existsP=> -[x y] /=; rewrite -/q -/r => /eqP[defG xq y2 xy].
have{} defG: <[x]> * <[y]> = G.
by rewrite -norm_joinEr // norms_cycle xy mem_cycle.
have notXy... | Lemma | generators_semidihedral | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Grp_semidihedral",
"TI_cardMg",
"apply",
"card_semidihedral",
"cycle_subG",
"def2qr",
"defG",
"double_inj",
"dvdn_leq",
"existsP",
"extremal_generators",
"group1_contra",
"inE",
"isoGrpP",
"isog",
"leqNgt",
"ltnW",
"ltqm",
"mem_cycle",
"mul2n",
"mulGSid",
"mulG_subl",
"m... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 => oG.
case/existsP=> -[x y] /=; rewrite -/q -/r => /eqP[defG xq y2 xy].
have{} defG: <[x]> * <[y]> = G.
by rewrite -norm_joinEr // norms_cycle xy groupV cycle_id.
have notXy:... | Lemma | generators_quaternion | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Grp_quaternion",
"apply",
"cardG_gt0",
"card_quaternion",
"cycle_id",
"cycle_subG",
"def2qr",
"defG",
"dvdn_leq",
"dvdn_mull",
"eqn_leq",
"existsP",
"expgM",
"extremal_generators",
"groupV",
"inE",
"isoGrpP",
"isog",
"last",
"leqNgt",
"leq_pmul2r",
"ltnW",
"ltqm",
"mem... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
X | := <[x]>. | Let | X | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Y | := <[y]>. | Let | Y | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
yG | := y ^: G. | Let | yG | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xyG | := (x * y) ^: G. | Let | xyG | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
My | := <<yG>>. | Let | My | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"yG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Mxy | := <<xyG>>. | Let | Mxy | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"xyG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 /\ (fo... | Proof.
move=> n_gt1 genG isoG; have [def2q def2r ltqm ltrq] := def2qr n_gt1.
have [oG Gx ox X'y] := genG; rewrite -/m -/q -/X in oG ox X'y.
case/extremal_generators_facts: genG; rewrite -/X // => pG maxX nsXG defXY nXY.
have [sXG nXG]:= andP nsXG; have [Gy notXy]:= setDP X'y.
have ox2: #[x ^+ 2] = r by rewrite orderXdi... | Theorem | dihedral2_structure | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Grp_2dihedral",
"Lagrange",
"Mho",
"MhoE",
"Mxy",
"My",
"OhmE",
"Ohm_p_cycle",
"Ohm_sub",
"Phi_joing",
"abelem",
"abelemP",
"addn0",
"addnn",
"apply",
"cGG",
"cardG_gt1",
"card_2dihedral",
"card_quotient",
"card_rcoset",
"cards0",
"cardsID",
"cardsUI",
"centP",
"cent... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 ^... | Proof.
move=> n_gt2 genG isoG; have [def2q def2r ltqm ltrq] := def2qr (ltnW n_gt2).
have [oG Gx ox X'y] := genG; rewrite -/m -/q -/X in oG ox X'y.
case/extremal_generators_facts: genG; rewrite -/X // => pG maxX nsXG defXY nXY.
have [sXG nXG]:= andP nsXG; have [Gy notXy]:= setDP X'y.
have oxr: #[x ^+ r] = 2 by rewrite o... | Theorem | quaternion_structure | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Grp_quaternion",
"Lagrange",
"Mho",
"MhoE",
"Mxy",
"My",
"OhmE",
"Ohm_p_cycle",
"Ohm_sub",
"Phi_joing",
"Phi_quotient_abelem",
"abelem_order_p",
"addn0",
"addnn",
"apply",
"cardG_gt1",
"card_quaternion",
"card_quotient",
"card_rcoset",
"cards0",
"cardsID",
"cardsUI",
"ce... | HERE BOOM | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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 ^+ ... | Proof.
move=> n_gt3 genG isoG oy.
have [def2q def2r ltqm ltrq] := def2qr (ltnW (ltnW n_gt3)).
have [oG Gx ox X'y] := genG; rewrite -/m -/q -/X in oG ox X'y.
case/extremal_generators_facts: genG; rewrite -/X // => pG maxX nsXG defXY nXY.
have [sXG nXG]:= andP nsXG; have [Gy notXy]:= setDP X'y.
have ox2: #[x ^+ 2] = r by... | Theorem | semidihedral_structure | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Gauss_dvdr",
"Grp_2dihedral",
"Grp_quaternion",
"Lagrange",
"Mho",
"MhoE",
"Mxy",
"My",
"OhmE",
"Ohm_p_cycle",
"Ohm_sub",
"Phi_joing",
"Phi_quotient_abelem",
"abelem_order_p",
"addn0",
"addnn",
"apply",
"cardG_gt1",
"card_2dihedral",
"card_quaternion",
"card_quotient",
"ca... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
extremal_group_type | :=
ModularGroup | Dihedral | SemiDihedral | Quaternion | NotExtremal. | Inductive | extremal_group_type | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
index_extremal_group_type c : nat | :=
match c with
| ModularGroup => 0
| Dihedral => 1
| SemiDihedral => 2
| Quaternion => 3
| NotExtremal => 4
end. | Definition | index_extremal_group_type | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
enum_extremal_groups | :=
[:: ModularGroup; Dihedral; SemiDihedral; Quaternion]. | Definition | enum_extremal_groups | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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