statement stringlengths 1 4.33k | proof stringlengths 0 37.9k | type stringclasses 25
values | symbolic_name stringlengths 1 67 | library stringclasses 10
values | filename stringclasses 112
values | imports listlengths 2 138 | deps listlengths 0 64 | docstring stringclasses 798
values | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
morphim_subnormal (rT : finGroupType) G (f : {morphism G >-> rT}) H K :
H <|<| K -> f @* H <|<| f @* K. | Proof.
case/subnormalP => s Hs <-{K}; apply/subnormalP.
elim: s H Hs => [|K s IHs] H /=; first by exists [::].
case/andP=> nsHK /IHs[fs Hfs <-].
by exists ([group of f @* K] :: fs); rewrite /= ?morphim_normal.
Qed. | Lemma | morphim_subnormal | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"apply",
"group",
"morphim_normal",
"morphism",
"subnormalP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quotient_subnormal H G K : G <|<| K -> G / H <|<| K / H. | Proof. exact: morphim_subnormal. Qed. | Lemma | quotient_subnormal | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"morphim_subnormal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
maximal_eqP M G :
reflect (M \subset G /\
forall H, M \subset H -> H \subset G -> H :=: M \/ H :=: G)
(maximal_eq M G). | Proof.
rewrite subEproper /maximal_eq; case: eqP => [->|_]; first left.
by split=> // H sGH sHG; right; apply/eqP; rewrite eqEsubset sHG.
apply: (iffP maxgroupP) => [] [sMG maxM]; split=> // H.
by move/maxM=> maxMH; rewrite subEproper; case/predU1P; auto.
by rewrite properEneq => /andP[/eqP neHG sHG] /maxM[].
Qed. | Lemma | maximal_eqP | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"apply",
"eqEsubset",
"maxgroupP",
"maximal_eq",
"predU1P",
"properEneq",
"sGH",
"sHG",
"split",
"subEproper"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
maximal_exists H G :
H \subset G ->
H :=: G \/ (exists2 M : {group gT}, maximal M G & H \subset M). | Proof.
rewrite subEproper; case/predU1P=> sHG; first by left.
suff [M *]: {M : {group gT} | maximal M G & H \subset M} by right; exists M.
exact: maxgroup_exists.
Qed. | Lemma | maximal_exists | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"gT",
"group",
"maxgroup_exists",
"maximal",
"predU1P",
"sHG",
"subEproper"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulg_normal_maximal G M H :
M <| G -> maximal M G -> H \subset G -> ~~ (H \subset M) -> (M * H = G)%g. | Proof.
case/andP=> sMG nMG /maxgroupP[_ maxM] sHG not_sHM.
apply/eqP; rewrite eqEproper mul_subG // -norm_joinEr ?(subset_trans sHG) //.
by apply: contra not_sHM => /maxM <-; rewrite ?joing_subl ?joing_subr.
Qed. | Lemma | mulg_normal_maximal | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"apply",
"eqEproper",
"joing_subl",
"joing_subr",
"maxgroupP",
"maximal",
"mul_subG",
"nMG",
"norm_joinEr",
"sHG",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
minnormal_exists G H : H :!=: 1 -> G \subset 'N(H) ->
{M : {group gT} | minnormal M G & M \subset H}. | Proof. by move=> ntH nHG; apply: mingroup_exists (H) _; rewrite ntH. Qed. | Lemma | minnormal_exists | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"apply",
"gT",
"group",
"mingroup_exists",
"minnormal",
"nHG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
(dM : M \subset f @* D) (dG : G \subset f @* D). | Hypotheses | dM | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
morphpre_maximal : maximal (f @*^-1 M) (f @*^-1 G) = maximal M G. | Proof.
apply/maxgroupP/maxgroupP; rewrite morphpre_proper //= => [] [ltMG maxM].
split=> // H ltHG sMH; have dH := subset_trans (proper_sub ltHG) dG.
rewrite -(morphpreK dH) [f @*^-1 H]maxM ?morphpreK ?morphpreSK //.
by rewrite morphpre_proper.
split=> // H ltHG sMH.
have dH: H \subset D := subset_trans (proper_s... | Lemma | morphpre_maximal | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"apply",
"dM",
"ker_sub_pre",
"maxgroupP",
"maximal",
"morphimGK",
"morphimS",
"morphpreK",
"morphpreSK",
"morphpre_proper",
"proper_sub",
"split",
"subsetIl",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphpre_maximal_eq : maximal_eq (f @*^-1 M) (f @*^-1 G) = maximal_eq M G. | Proof. by rewrite /maximal_eq morphpre_maximal !eqEsubset !morphpreSK. Qed. | Lemma | morphpre_maximal_eq | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"eqEsubset",
"maximal_eq",
"morphpreSK",
"morphpre_maximal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
(dM : M \subset D) (dG : G \subset D) (dL : L \subset D). | Hypotheses | dM | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
injm_maximal : maximal (f @* M) (f @* G) = maximal M G. | Proof.
rewrite -(morphpre_invm injf) -(morphpre_invm injf G).
by rewrite morphpre_maximal ?morphim_invm.
Qed. | Lemma | injm_maximal | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"injf",
"maximal",
"morphim_invm",
"morphpre_invm",
"morphpre_maximal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
injm_maximal_eq : maximal_eq (f @* M) (f @* G) = maximal_eq M G. | Proof. by rewrite /maximal_eq injm_maximal // injm_eq. Qed. | Lemma | injm_maximal_eq | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"injm_eq",
"injm_maximal",
"maximal_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
injm_maxnormal : maxnormal (f @* M) (f @* G) (f @* L) = maxnormal M G L. | Proof.
pose injfm := (injm_proper injf, injm_norms, injmSK injf, subsetIl).
apply/maxgroupP/maxgroupP; rewrite !injfm // => [[nML maxM]].
split=> // H nHL sMH; have [/proper_sub sHG _] := andP nHL.
have dH := subset_trans sHG dG; apply: (injm_morphim_inj injf) => //.
by apply: maxM; rewrite !injfm.
split=> // fH ... | Lemma | injm_maxnormal | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"apply",
"fH",
"injf",
"injmSK",
"injm_morphim_inj",
"injm_norms",
"injm_proper",
"maxgroupP",
"maxnormal",
"morphim_sub",
"morphpreK",
"proper_sub",
"sHG",
"split",
"subsetIl",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
injm_minnormal : minnormal (f @* M) (f @* G) = minnormal M G. | Proof.
pose injfm := (morphim_injm_eq1 injf, injm_norms, injmSK injf, subsetIl).
apply/mingroupP/mingroupP; rewrite !injfm // => [[nML minM]].
split=> // H nHG sHM; have dH := subset_trans sHM dM.
by apply: (injm_morphim_inj injf) => //; apply: minM; rewrite !injfm.
split=> // fH nHG sHM; have dfH := subset_trans s... | Lemma | injm_minnormal | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"apply",
"dM",
"fH",
"injf",
"injmSK",
"injm_morphim_inj",
"injm_norms",
"mingroupP",
"minnormal",
"morphim_injm_eq1",
"morphim_sub",
"morphpreK",
"nHG",
"split",
"subsetIl",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cosetpre_maximal (Q R : {group coset_of K}) :
maximal (coset K @*^-1 Q) (coset K @*^-1 R) = maximal Q R. | Proof. by rewrite morphpre_maximal ?sub_im_coset. Qed. | Lemma | cosetpre_maximal | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"coset",
"coset_of",
"group",
"maximal",
"morphpre_maximal",
"sub_im_coset"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cosetpre_maximal_eq (Q R : {group coset_of K}) :
maximal_eq (coset K @*^-1 Q) (coset K @*^-1 R) = maximal_eq Q R. | Proof. by rewrite /maximal_eq !eqEsubset !cosetpreSK cosetpre_maximal. Qed. | Lemma | cosetpre_maximal_eq | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"coset",
"coset_of",
"cosetpreSK",
"cosetpre_maximal",
"eqEsubset",
"group",
"maximal_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quotient_maximal :
K <| G -> K <| H -> maximal (G / K) (H / K) = maximal G H. | Proof. by move=> nKG nKH; rewrite -cosetpre_maximal ?quotientGK. Qed. | Lemma | quotient_maximal | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"cosetpre_maximal",
"maximal",
"nKG",
"nKH",
"quotientGK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quotient_maximal_eq :
K <| G -> K <| H -> maximal_eq (G / K) (H / K) = maximal_eq G H. | Proof. by move=> nKG nKH; rewrite -cosetpre_maximal_eq ?quotientGK. Qed. | Lemma | quotient_maximal_eq | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"cosetpre_maximal_eq",
"maximal_eq",
"nKG",
"nKH",
"quotientGK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
maximalJ x : maximal (G :^ x) (H :^ x) = maximal G H. | Proof.
rewrite -{1}(setTI G) -{1}(setTI H) -!morphim_conj.
by rewrite injm_maximal ?subsetT ?injm_conj.
Qed. | Lemma | maximalJ | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"injm_conj",
"injm_maximal",
"maximal",
"morphim_conj",
"setTI",
"subsetT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
maximal_eqJ x : maximal_eq (G :^ x) (H :^ x) = maximal_eq G H. | Proof. by rewrite /maximal_eq !eqEsubset !conjSg maximalJ. Qed. | Lemma | maximal_eqJ | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"conjSg",
"eqEsubset",
"maximalJ",
"maximal_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
maxnormal_normal A B : maxnormal A B B -> A <| B. | Proof.
by case/maxsetP=> /and3P[/gen_set_id /= -> pAB nAB]; rewrite /normal proper_sub.
Qed. | Lemma | maxnormal_normal | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"gen_set_id",
"maxnormal",
"maxsetP",
"normal",
"proper_sub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
maxnormal_proper A B C : maxnormal A B C -> A \proper B. | Proof.
by case/maxsetP=> /and3P[gA pAB _] _; apply: (sub_proper_trans (subset_gen A)).
Qed. | Lemma | maxnormal_proper | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"apply",
"maxnormal",
"maxsetP",
"proper",
"sub_proper_trans",
"subset_gen"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
maxnormal_sub A B C : maxnormal A B C -> A \subset B. | Proof.
by move=> maxA; rewrite proper_sub //; apply: (maxnormal_proper maxA).
Qed. | Lemma | maxnormal_sub | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"apply",
"maxA",
"maxnormal",
"maxnormal_proper",
"proper_sub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ex_maxnormal_ntrivg G : G :!=: 1-> {N : {group gT} | maxnormal N G G}. | Proof.
move=> ntG; apply: ex_maxgroup; exists [1 gT]%G; rewrite norm1 proper1G.
by rewrite subsetT ntG.
Qed. | Lemma | ex_maxnormal_ntrivg | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"apply",
"ex_maxgroup",
"gT",
"group",
"maxnormal",
"norm1",
"proper1G",
"subsetT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
maxnormalM G H K :
maxnormal H G G -> maxnormal K G G -> H :<>: K -> H * K = G. | Proof.
move=> maxH maxK /eqP; apply: contraNeq => ltHK_G.
have [nsHG nsKG] := (maxnormal_normal maxH, maxnormal_normal maxK).
have cHK: commute H K.
exact: normC (subset_trans (normal_sub nsHG) (normal_norm nsKG)).
wlog suffices: H K {maxH} maxK nsHG nsKG cHK ltHK_G / H \subset K.
by move=> IH; rewrite eqEsubset !I... | Lemma | maxnormalM | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"apply",
"comm_joingE",
"commute",
"contraNeq",
"eqEsubset",
"joing_idPr",
"joing_subr",
"maxgroupP",
"maxnormal",
"maxnormal_normal",
"normC",
"normalM",
"normal_norm",
"normal_sub",
"nsHG",
"nsKG",
"properEneq",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
maxnormal_minnormal G L M :
G \subset 'N(M) -> L \subset 'N(G) -> maxnormal M G L ->
minnormal (G / M) (L / M). | Proof.
move=> nMG nGL /maxgroupP[/andP[/andP[sMG ltMG] nML] maxM]; apply/mingroupP.
rewrite -subG1 quotient_sub1 ?ltMG ?quotient_norms //.
split=> // Hb /andP[ntHb nHbL]; have nsMG: M <| G by apply/andP.
case/inv_quotientS=> // H defHb sMH sHG; rewrite defHb; congr (_ / M).
apply/eqP; rewrite eqEproper sHG /=; apply: c... | Lemma | maxnormal_minnormal | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"apply",
"eqEproper",
"inv_quotientS",
"maxgroupP",
"maxnormal",
"mingroupP",
"minnormal",
"nMG",
"norm_quotient_pre",
"normalS",
"quotientGK",
"quotientS1",
"quotient_norms",
"quotient_sub1",
"sHG",
"split",
"subG1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
minnormal_maxnormal G L M :
M <| G -> L \subset 'N(M) -> minnormal (G / M) (L / M) -> maxnormal M G L. | Proof.
case/andP=> sMG nMG nML /mingroupP[/andP[/= ntGM _] minGM]; apply/maxgroupP.
split=> [|H /andP[/andP[sHG ltHG] nHL] sMH].
by rewrite /proper sMG nML andbT; apply: contra ntGM => /quotientS1 ->.
apply/eqP; rewrite eqEsubset sMH andbT -quotient_sub1 ?(subset_trans sHG) //.
rewrite subG1; apply: contraR ltHG => n... | Lemma | minnormal_maxnormal | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"apply",
"eqEsubset",
"maxgroupP",
"maxnormal",
"mingroupP",
"minnormal",
"nMG",
"proper",
"quotientS",
"quotientS1",
"quotientSGK",
"quotient_norms",
"quotient_sub1",
"sHG",
"split",
"subG1",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
simpleP gT (G : {group gT}) :
reflect (G :!=: 1 /\ forall H : {group gT}, H <| G -> H :=: 1 \/ H :=: G)
(simple G). | Proof.
apply: (iffP mingroupP); rewrite normG andbT => [[ntG simG]].
split=> // N /andP[sNG nNG].
by case: (eqsVneq N 1) => [|ntN]; [left | right; apply: simG; rewrite ?ntN].
split=> // N /andP[ntN nNG] sNG.
by case: (simG N) ntN => // [|->]; [apply/andP | case/eqP].
Qed. | Lemma | simpleP | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"apply",
"eqsVneq",
"gT",
"group",
"mingroupP",
"nNG",
"normG",
"sNG",
"simple",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quotient_simple gT (G H : {group gT}) :
H <| G -> simple (G / H) = maxnormal H G G. | Proof.
move=> nsHG; have nGH := normal_norm nsHG.
by apply/idP/idP; [apply: minnormal_maxnormal | apply: maxnormal_minnormal].
Qed. | Lemma | quotient_simple | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"apply",
"gT",
"group",
"maxnormal",
"maxnormal_minnormal",
"minnormal_maxnormal",
"normal_norm",
"nsHG",
"simple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isog_simple gT rT (G : {group gT}) (M : {group rT}) :
G \isog M -> simple G = simple M. | Proof.
move=> eqGM; wlog suffices: gT rT G M eqGM / simple M -> simple G.
by move=> IH; apply/idP/idP; apply: IH; rewrite // isog_sym.
case/isogP: eqGM => f injf <- /simpleP[ntGf simGf].
apply/simpleP; split=> [|N nsNG]; first by rewrite -(morphim_injm_eq1 injf).
rewrite -(morphim_invm injf (normal_sub nsNG)).
have: ... | Lemma | isog_simple | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"apply",
"gT",
"group",
"injf",
"isog",
"isogP",
"isog_sym",
"morphim1",
"morphim_injm_eq1",
"morphim_invm",
"morphim_normal",
"normal_sub",
"nsNG",
"simple",
"simpleP",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
simple_maxnormal gT (G : {group gT}) : simple G = maxnormal 1 G G. | Proof.
by rewrite -quotient_simple ?normal1 // -(isog_simple (quotient1_isog G)).
Qed. | Lemma | simple_maxnormal | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"gT",
"group",
"isog_simple",
"maxnormal",
"normal1",
"quotient1_isog",
"quotient_simple",
"simple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
chief_factor_minnormal G V U :
chief_factor G V U -> minnormal (U / V) (G / V). | Proof.
case/andP=> maxV /andP[sUG nUG]; apply: maxnormal_minnormal => //.
by have /andP[_ nVG] := maxgroupp maxV; apply: subset_trans sUG nVG.
Qed. | Lemma | chief_factor_minnormal | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"apply",
"chief_factor",
"maxgroupp",
"maxnormal_minnormal",
"minnormal",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
acts_irrQ G U V :
G \subset 'N(V) -> V <| U ->
acts_irreducibly G (U / V) 'Q = minnormal (U / V) (G / V). | Proof.
move=> nVG nsVU; apply/mingroupP/mingroupP; case=> /andP[->] /=.
rewrite astabsQ // subsetI nVG /= => nUG minUV.
rewrite quotient_norms //; split=> // H /andP[ntH nHG] sHU.
by apply: minUV (sHU); rewrite ntH -(cosetpreK H) actsQ // norm_quotient_pre.
rewrite sub_quotient_pre // => nUG minU; rewrite astabsQ... | Lemma | acts_irrQ | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"actsQ",
"acts_irreducibly",
"apply",
"astabsQ",
"cosetpreK",
"mingroupP",
"minnormal",
"morphpre_norm",
"nHG",
"norm_quotient_pre",
"normal_cosetpre",
"quotientGK",
"quotientS",
"quotient_norm",
"quotient_norms",
"split",
"sub_quotient_pre",
"subsetI",
"subsetIl",
"subset_tran... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
chief_series_exists H G :
H <| G -> {s | (G.-chief).-series 1%G s & last 1%G s = H}. | Proof.
have [m] := ubnP #|H|; elim: m H => // m IHm U leUm nsUG.
have [-> | ntU] := eqVneq U 1%G; first by exists [::].
have [V maxV]: {V : {group gT} | maxnormal V U G}.
by apply: ex_maxgroup; exists 1%G; rewrite proper1G ntU norms1.
have /andP[ltVU nVG] := maxgroupp maxV.
have [||s ch_s defV] := IHm V; first exact:... | Lemma | chief_series_exists | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"apply",
"chief_factor",
"eqVneq",
"ex_maxgroup",
"gT",
"group",
"last",
"last_rcons",
"leq_trans",
"maxgroupp",
"maxnormal",
"normal",
"normal_sub",
"norms1",
"proper1G",
"proper_card",
"proper_sub",
"rcons",
"rcons_path",
"subset_trans",
"ubnP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
central_factor_central H K :
central_factor G H K -> (K / H) \subset 'Z(G / H). | Proof. by case/and3P=> /quotient_cents2r *; rewrite subsetI quotientS. Qed. | Lemma | central_factor_central | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"central_factor",
"quotientS",
"quotient_cents2r",
"subsetI"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
central_central_factor H K :
(K / H) \subset 'Z(G / H) -> H <| K -> H <| G -> central_factor G H K. | Proof.
case/subsetIP=> sKGb cGKb /andP[sHK nHK] /andP[sHG nHG].
by rewrite /central_factor -quotient_cents2 // cGKb sHK -(quotientSGK nHK).
Qed. | Lemma | central_central_factor | solvable | solvable/gseries.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"commutator",
"center"
] | [
"central_factor",
"nHG",
"nHK",
"quotientSGK",
"quotient_cents2",
"sHG",
"sHK",
"subsetIP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
SchurZassenhaus_split gT (G H : {group gT}) :
Hall G H -> H <| G -> [splits G, over H]. | Proof.
have [n] := ubnP #|G|; elim: n => // n IHn in gT G H * => /ltnSE-Gn hallH nsHG.
have [sHG nHG] := andP nsHG.
have [-> | [p pr_p pH]] := trivgVpdiv H.
by apply/splitsP; exists G; rewrite inE -subG1 subsetIl mul1g eqxx.
have [P sylP] := Sylow_exists p H.
case nPG: (P <| G); last first.
pose N := ('N_G(P))%G; h... | Theorem | SchurZassenhaus_split | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"Frattini_arg",
"Gaschutz_split",
"Hall",
"Lagrange",
"Sylow_exists",
"TI_cardMg",
"apply",
"cardG_gt0",
"card_Hall",
"card_quotient",
"center_abelian",
"center_sub",
"complP",
"coprimeSg",
"divgS",
"divnMl",
"eqEcard",
"eqEsubset",
"eqn_exp2l",
"eqxx",
"expn0",
"gFnormal_t... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
SchurZassenhaus_trans_sol gT (H K K1 : {group gT}) :
solvable H -> K \subset 'N(H) -> K1 \subset H * K ->
coprime #|H| #|K| -> #|K1| = #|K| ->
exists2 x, x \in H & K1 :=: K :^ x. | Proof.
have [n] := ubnP #|H|.
elim: n => // n IHn in gT H K K1 * => /ltnSE-leHn solH nHK.
have [-> | ] := eqsVneq H 1.
rewrite mul1g => sK1K _ eqK1K; exists 1; first exact: set11.
by apply/eqP; rewrite conjsg1 eqEcard sK1K eqK1K /=.
pose G := (H <*> K)%G.
have defG: G :=: H * K by rewrite -normC // -norm_joinEl // ... | Theorem | SchurZassenhaus_trans_sol | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"Gaschutz_transitive",
"Lagrange",
"My",
"apply",
"cardJg",
"card_quotient",
"conjsg1",
"conjsgM",
"coprime",
"coprimeMl",
"coprime_TIg",
"coprime_cardMg",
"coprime_sym",
"defG",
"divgS",
"eqEcard",
"eq_sym",
"eqsVneq",
"eqxx",
"gT",
"genS",
"gen_subG",
"group",
"groupM... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
SchurZassenhaus_trans_actsol gT (G A B : {group gT}) :
solvable A -> A \subset 'N(G) -> B \subset A <*> G ->
coprime #|G| #|A| -> #|A| = #|B| ->
exists2 x, x \in G & B :=: A :^ x. | Proof.
set AG := A <*> G; have [n] := ubnP #|AG|.
elim: n => // n IHn in gT A B G AG * => /ltnSE-leAn solA nGA sB_AG coGA oAB.
have [A1 | ntA] := eqsVneq A 1.
by exists 1; rewrite // conjsg1 A1 (@card1_trivg _ B) // -oAB A1 cards1.
have [M [sMA nsMA ntM]] := solvable_norm_abelem solA (normal_refl A) ntA.
case/is_abel... | Lemma | SchurZassenhaus_trans_actsol | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"Hall_max",
"Sylow",
"Sylow_exists",
"Sylow_trans",
"abelem_pgroup",
"apply",
"card1_trivg",
"cardJg",
"card_Hall",
"card_quotient",
"cards1",
"coGA",
"conjGid",
"conjSg",
"conjsg1",
"conjsgKV",
"conjsgM",
"coprime",
"coprimeSg",
"coprime_cardMg",
"coprime_morph",
"coprime_... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Hall_exists_subJ pi gT (G : {group gT}) :
solvable G -> exists2 H : {group gT}, pi.-Hall(G) H
& forall K : {group gT}, K \subset G -> pi.-group K ->
exists2 x, x \in G & K \subset H :^ x. | Proof.
have [n] := ubnP #|G|; elim: n gT G => // n IHn gT G /ltnSE-leGn solG.
have [-> | ntG] := eqsVneq G 1.
exists 1%G => [|_ /trivGP-> _]; last by exists 1; rewrite ?set11 ?sub1G.
by rewrite pHallE sub1G cards1 part_p'nat.
case: (solvable_norm_abelem solG (normal_refl _)) => // M [sMG nsMG ntM].
case/is_abelemP=... | Lemma | Hall_exists_subJ | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"G1",
"Hall",
"Lagrange",
"SchurZassenhaus_split",
"SchurZassenhaus_trans_sol",
"TI_cardMg",
"apply",
"cardG_gt0",
"cardJg",
"card_quotient",
"cards1",
"complP",
"conjSg",
"conjsgK",
"conjsgKV",
"conjsgM",
"coprime",
"coprime_cardMg",
"coprime_sym",
"coset",
"defG",
"divgS"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Hall_exists pi (G : {group gT}) :
solvable G -> exists H : {group gT}, pi.-Hall(G) H. | Proof. by case/(Hall_exists_subJ pi) => H; exists H. Qed. | Corollary | Hall_exists | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"Hall",
"Hall_exists_subJ",
"gT",
"group",
"pi",
"solvable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Hall_trans pi (G H1 H2 : {group gT}) :
solvable G -> pi.-Hall(G) H1 -> pi.-Hall(G) H2 ->
exists2 x, x \in G & H1 :=: H2 :^ x. | Proof.
move=> solG; have [H hallH transH] := Hall_exists_subJ pi solG.
have conjH (K : {group gT}):
pi.-Hall(G) K -> exists2 x, x \in G & K = (H :^ x)%G.
- move=> hallK; have [sKG piK _] := and3P hallK.
case: (transH K sKG piK) => x Gx sKH; exists x => //.
apply/eqP; rewrite -val_eqE eqEcard sKH cardJg.
by rewr... | Corollary | Hall_trans | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"Hall",
"Hall_exists_subJ",
"apply",
"cardJg",
"card_Hall",
"conjsgK",
"conjsgM",
"eqEcard",
"gT",
"group",
"groupMl",
"groupV",
"pi",
"piK",
"sKG",
"solvable",
"val_eqE",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Hall_superset pi (G K : {group gT}) :
solvable G -> K \subset G -> pi.-group K ->
exists2 H : {group gT}, pi.-Hall(G) H & K \subset H. | Proof.
move=> solG sKG; have [H hallH transH] := Hall_exists_subJ pi solG.
by case/transH=> // x Gx sKHx; exists (H :^ x)%G; rewrite ?pHallJ.
Qed. | Corollary | Hall_superset | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"Hall",
"Hall_exists_subJ",
"gT",
"group",
"pHallJ",
"pi",
"sKG",
"solvable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Hall_subJ pi (G H K : {group gT}) :
solvable G -> pi.-Hall(G) H -> K \subset G -> pi.-group K ->
exists2 x, x \in G & K \subset H :^ x. | Proof.
move=> solG HallH sKG piK; have [M HallM sKM]:= Hall_superset solG sKG piK.
have [x Gx defM] := Hall_trans solG HallM HallH.
by exists x; rewrite // -defM.
Qed. | Corollary | Hall_subJ | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"Hall",
"Hall_superset",
"Hall_trans",
"gT",
"group",
"pi",
"piK",
"sKG",
"solvable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Hall_Jsub pi (G H K : {group gT}) :
solvable G -> pi.-Hall(G) H -> K \subset G -> pi.-group K ->
exists2 x, x \in G & K :^ x \subset H. | Proof.
move=> solG HallH sKG piK; have [x Gx sKHx] := Hall_subJ solG HallH sKG piK.
by exists x^-1; rewrite ?groupV // sub_conjgV.
Qed. | Corollary | Hall_Jsub | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"Hall",
"Hall_subJ",
"gT",
"group",
"groupV",
"pi",
"piK",
"sKG",
"solvable",
"sub_conjgV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Hall_Frattini_arg pi (G K H : {group gT}) :
solvable K -> K <| G -> pi.-Hall(K) H -> K * 'N_G(H) = G. | Proof.
move=> solK /andP[sKG nKG] hallH.
have sHG: H \subset G by apply: subset_trans sKG; case/andP: hallH.
rewrite setIC group_modl //; apply/setIidPr/subsetP=> x Gx.
pose H1 := (H :^ x^-1)%G.
have hallH1: pi.-Hall(K) H1 by rewrite pHallJnorm // groupV (subsetP nKG).
case: (Hall_trans solK hallH hallH1) => y Ky defH.... | Lemma | Hall_Frattini_arg | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"Hall",
"Hall_trans",
"apply",
"conjsgK",
"conjsgKV",
"conjsgM",
"gT",
"group",
"groupV",
"group_modl",
"mem_mulg",
"mulKVg",
"nKG",
"normP",
"pHallJnorm",
"pi",
"sHG",
"sKG",
"setIC",
"setIidPr",
"solvable",
"subsetP",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprime_norm_cent A G :
A \subset 'N(G) -> coprime #|G| #|A| -> 'N_G(A) = 'C_G(A). | Proof.
move=> nGA coGA; apply/eqP; rewrite eqEsubset andbC setIS ?cent_sub //=.
rewrite subsetI subsetIl /= (sameP commG1P trivgP) -(coprime_TIg coGA).
rewrite subsetI commg_subr subsetIr andbT.
move: nGA; rewrite -commg_subl; apply: subset_trans.
by rewrite commSg ?subsetIl.
Qed. | Lemma | coprime_norm_cent | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"apply",
"cent_sub",
"coGA",
"commG1P",
"commSg",
"commg_subl",
"commg_subr",
"coprime",
"coprime_TIg",
"eqEsubset",
"setIS",
"subsetI",
"subsetIl",
"subsetIr",
"subset_trans",
"trivgP"
] | Part of Aschbacher (18.7.4). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
coprime_Hall_exists A G :
A \subset 'N(G) -> coprime #|G| #|A| -> solvable G ->
exists2 H : {group gT}, pi.-Hall(G) H & A \subset 'N(H). | Proof.
move=> nGA coGA solG; have [H hallH] := Hall_exists pi solG.
have sG_AG: G \subset A <*> G by rewrite joing_subr.
have nG_AG: A <*> G \subset 'N(G) by rewrite join_subG nGA normG.
pose N := 'N_(A <*> G)(H)%G.
have nGN: N \subset 'N(G) by rewrite subIset ?nG_AG.
have nGN_N: G :&: N <| N by rewrite /(_ <| N) subse... | Proposition | coprime_Hall_exists | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"Hall",
"Hall_Frattini_arg",
"Hall_exists",
"SchurZassenhaus_split",
"SchurZassenhaus_trans_sol",
"TI_cardMg",
"apply",
"card_quotient",
"coGA",
"complP",
"conjgCV",
"conjsgM",
"coprime",
"coprimeSg",
"coprime_cardMg",
"coprime_sym",
"divgI",
"divgS",
"gT",
"group",
"groupV",... | This is B & G, Proposition 1.5(a) | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
coprime_Hall_trans A G H1 H2 :
A \subset 'N(G) -> coprime #|G| #|A| -> solvable G ->
pi.-Hall(G) H1 -> A \subset 'N(H1) ->
pi.-Hall(G) H2 -> A \subset 'N(H2) ->
exists2 x, x \in 'C_G(A) & H1 :=: H2 :^ x. | Proof.
move: H1 => H nGA coGA solG hallH nHA hallH2.
have{H2 hallH2} [x Gx -> nH1xA] := Hall_trans solG hallH2 hallH.
have sG_AG: G \subset A <*> G by rewrite -{1}genGid genS ?subsetUr.
have nG_AG: A <*> G \subset 'N(G) by rewrite gen_subG subUset nGA normG.
pose N := 'N_(A <*> G)(H)%G.
have nGN: N \subset 'N(G) by rew... | Proposition | coprime_Hall_trans | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"Hall",
"Hall_Frattini_arg",
"Hall_trans",
"Lagrange",
"SchurZassenhaus_trans_sol",
"apply",
"cardJg",
"card_isog",
"card_quotient",
"coGA",
"conjGid",
"conjIg",
"conjsgK",
"conjsgKV",
"conjsgM",
"coprime",
"coprimeSg",
"coprime_cardMg",
"coprime_norm_cent",
"divgS",
"eqEcard... | This is B & G, Proposition 1.5(c) | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
norm_conj_cent A G x : x \in 'C(A) ->
(A \subset 'N(G :^ x)) = (A \subset 'N(G)). | Proof. by move=> cAx; rewrite norm_conj_norm ?(subsetP (cent_sub A)). Qed. | Lemma | norm_conj_cent | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"cent_sub",
"norm_conj_norm",
"subsetP"
] | A complement to the above: 'C(A) acts on 'Nby(A) | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
strongest_coprime_quotient_cent A G H :
let R := H :&: [~: G, A] in
A \subset 'N(H) -> R \subset G -> coprime #|R| #|A| ->
solvable R || solvable A ->
'C_G(A) / H = 'C_(G / H)(A / H). | Proof.
move=> R nHA sRG coRA solRA.
have nRA: A \subset 'N(R) by rewrite normsI ?commg_normr.
apply/eqP; rewrite eqEsubset subsetI morphimS ?subsetIl //=.
rewrite (subset_trans _ (morphim_cent _ _)) ?morphimS ?subsetIr //=.
apply/subsetP=> _ /setIP[/morphimP[x Nx Gx ->] cAHx].
have{cAHx} cAxR y: y \in A -> [~ x, y] \in... | Lemma | strongest_coprime_quotient_cent | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"SchurZassenhaus_trans_actsol",
"SchurZassenhaus_trans_sol",
"apply",
"cardJg",
"centP",
"commgEl",
"commgEr",
"commgP",
"commg_normr",
"conjMg",
"conjVg",
"conjgCV",
"conjgM",
"coprime",
"coprime_TIg",
"coset_idr",
"eqEsubset",
"groupMl",
"groupMr",
"groupR",
"groupV",
"im... | Odd Order Theorem. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
coprime_norm_quotient_cent A G H :
A \subset 'N(G) -> A \subset 'N(H) -> coprime #|H| #|A| -> solvable H ->
'C_G(A) / H = 'C_(G / H)(A / H). | Proof.
move=> nGA nHA coHA solH; have sRH := subsetIl H [~: G, A].
rewrite strongest_coprime_quotient_cent ?(coprimeSg sRH) 1?(solvableS sRH) //.
by rewrite subIset // commg_subl nGA orbT.
Qed. | Lemma | coprime_norm_quotient_cent | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"commg_subl",
"coprime",
"coprimeSg",
"solvable",
"solvableS",
"strongest_coprime_quotient_cent",
"subIset",
"subsetIl"
] | needed in this case. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
coprime_cent_mulG A G H :
A \subset 'N(G) -> A \subset 'N(H) -> G \subset 'N(H) ->
coprime #|H| #|A| -> solvable H ->
'C_(H * G)(A) = 'C_H(A) * 'C_G(A). | Proof.
move=> nHA nGA nHG coHA solH; rewrite -norm_joinEr //.
have nsHG: H <| H <*> G by rewrite /normal joing_subl join_subG normG.
rewrite -{2}(setIidPr (normal_sub nsHG)) setIAC.
rewrite group_modr ?setSI ?joing_subr //=; symmetry; apply/setIidPl.
rewrite -quotientSK ?subIset 1?normal_norm //.
by rewrite !coprime_no... | Lemma | coprime_cent_mulG | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"apply",
"coprime",
"coprime_norm_quotient_cent",
"group_modr",
"join_subG",
"joing_subl",
"joing_subr",
"nHG",
"normG",
"norm_joinEr",
"normal",
"normal_norm",
"normal_sub",
"normsY",
"nsHG",
"quotientMidl",
"quotientSK",
"setIAC",
"setIidPl",
"setIidPr",
"setSI",
"solvabl... | theorem. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
quotient_TI_subcent K G H :
G \subset 'N(K) -> G \subset 'N(H) -> K :&: H = 1 ->
'C_K(G) / H = 'C_(K / H)(G / H). | Proof.
move=> nGK nGH tiKH.
have tiHR: H :&: [~: K, G] = 1.
by apply/trivgP; rewrite /= setIC -tiKH setSI ?commg_subl.
apply: strongest_coprime_quotient_cent; rewrite ?tiHR ?sub1G ?solvable1 //.
by rewrite cards1 coprime1n.
Qed. | Lemma | quotient_TI_subcent | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"apply",
"cards1",
"commg_subl",
"coprime1n",
"setIC",
"setSI",
"solvable1",
"strongest_coprime_quotient_cent",
"sub1G",
"tiKH",
"trivgP"
] | justified by switching to external action. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
coprime_quotient_cent A G H :
H \subset G -> A \subset 'N(H) -> coprime #|G| #|A| -> solvable G ->
'C_G(A) / H = 'C_(G / H)(A / H). | Proof.
move=> sHG nHA coGA solG.
have sRG: H :&: [~: G, A] \subset G by rewrite subIset ?sHG.
by rewrite strongest_coprime_quotient_cent ?(coprimeSg sRG) 1?(solvableS sRG).
Qed. | Proposition | coprime_quotient_cent | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"coGA",
"coprime",
"coprimeSg",
"sHG",
"solvable",
"solvableS",
"strongest_coprime_quotient_cent",
"subIset"
] | coprime and solvability assumptions are easier to satisfy in practice. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
coprime_comm_pcore A G K :
A \subset 'N(G) -> coprime #|G| #|A| -> solvable G ->
pi^'.-Hall(G) K -> K \subset 'C_G(A) ->
[~: G, A] \subset 'O_pi(G). | Proof.
move=> nGA coGA solG hallK cKA.
case: (coprime_Hall_exists nGA) => // H hallH nHA.
have sHG: H \subset G by case/andP: hallH.
have sKG: K \subset G by case/andP: hallK.
have coKH: coprime #|K| #|H|.
case/and3P: hallH=> _ piH _; case/and3P: hallK => _ pi'K _.
by rewrite coprime_sym (pnat_coprime piH pi'K).
ha... | Proposition | coprime_comm_pcore | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"Hall",
"apply",
"card_Hall",
"centsP",
"coGA",
"commGC",
"commMgJ",
"commgP",
"commg_norml",
"commg_subr",
"conj1g",
"coprime",
"coprime_Hall_exists",
"coprime_cardMg",
"coprime_sym",
"defG",
"eqEcard",
"eq_sym",
"gen_subG",
"groupMl",
"groupV",
"imset2P",
"last",
"mem... | This is B & G, Proposition 1.5(e). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
coprime_Hall_subset pi (gT : finGroupType) (A G X : {group gT}) :
A \subset 'N(G) -> coprime #|G| #|A| -> solvable G ->
X \subset G -> pi.-group X -> A \subset 'N(X) ->
exists H : {group gT}, [/\ pi.-Hall(G) H, A \subset 'N(H) & X \subset H]. | Proof.
have [n] := ubnP #|G|.
elim: n => // n IHn in gT A G X * => /ltnSE-leGn nGA coGA solG sXG piX nXA.
have [G1 | ntG] := eqsVneq G 1.
case: (coprime_Hall_exists pi nGA) => // H hallH nHA.
by exists H; split; rewrite // (subset_trans sXG) // G1 sub1G.
have sG_AG: G \subset A <*> G by rewrite joing_subr.
have sA_... | Proposition | coprime_Hall_subset | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"G1",
"Hall",
"Lagrange",
"apply",
"cardG_gt0",
"card_Hall",
"card_quotient",
"coGA",
"conjSg",
"coprime",
"coprimeSg",
"coprime_Hall_exists",
"coprime_Hall_trans",
"coprime_cardMg",
"coprime_morph",
"divgS",
"divnMl",
"eqEcard",
"eqsVneq",
"gT",
"group",
"group_modr",
"i... | This is B & G, Proposition 1.5(b). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
inA | := (sdpair2 to). | Notation | inA | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"sdpair2",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inG | := (sdpair1 to). | Notation | inG | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"sdpair1",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
A' | := (inA @* gval A). | Notation | A' | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"inA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
G' | := (inG @* gval G). | Notation | G' | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"inG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
injG : 'injm inG | := injm_sdpair1 _. | Let | injG | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"inG",
"injm_sdpair1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
injA : 'injm inA | := injm_sdpair2 _. | Let | injA | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"inA",
"injm_sdpair2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
(coGA : coprime #|G| #|A|) (solG : solvable G). | Hypotheses | coGA | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"coprime",
"solvable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
external_action_im_coprime : coprime #|G'| #|A'|. | Proof. by rewrite !card_injm. Qed. | Lemma | external_action_im_coprime | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"A'",
"G'",
"card_injm",
"coprime"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coGA' | := external_action_im_coprime. | Let | coGA' | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"external_action_im_coprime"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
solG' : solvable G' | := morphim_sol _ solG. | Let | solG' | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"G'",
"morphim_sol",
"solvable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nGA' | := im_sdpair_norm to. | Let | nGA' | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"im_sdpair_norm",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ext_coprime_Hall_exists :
exists2 H : {group gT}, pi.-Hall(G) H & [acts A, on H | to]. | Proof.
have [H' hallH' nHA'] := coprime_Hall_exists pi nGA' coGA' solG'.
have sHG' := pHall_sub hallH'.
exists (inG @*^-1 H')%G => /=.
by rewrite -(morphim_invmE injG) -{1}(im_invm injG) morphim_pHall.
by rewrite actsEsd ?morphpreK // subsetIl.
Qed. | Lemma | ext_coprime_Hall_exists | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"Hall",
"actsEsd",
"coGA'",
"coprime_Hall_exists",
"gT",
"group",
"im_invm",
"inG",
"injG",
"morphim_invmE",
"morphim_pHall",
"morphpreK",
"nGA'",
"on",
"pHall_sub",
"pi",
"solG'",
"subsetIl",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ext_coprime_Hall_trans (H1 H2 : {group gT}) :
pi.-Hall(G) H1 -> [acts A, on H1 | to] ->
pi.-Hall(G) H2 -> [acts A, on H2 | to] ->
exists2 x, x \in 'C_(G | to)(A) & H1 :=: H2 :^ x. | Proof.
move=> hallH1 nH1A hallH2 nH2A.
have sH1G := pHall_sub hallH1; have sH2G := pHall_sub hallH2.
rewrite !actsEsd // in nH1A nH2A.
have hallH1': pi.-Hall(G') (inG @* H1) by rewrite morphim_pHall.
have hallH2': pi.-Hall(G') (inG @* H2) by rewrite morphim_pHall.
have [x'] := coprime_Hall_trans nGA' coGA' solG' hallH1... | Lemma | ext_coprime_Hall_trans | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"G'",
"Hall",
"actsEsd",
"apply",
"coGA'",
"conj_subG",
"coprime_Hall_trans",
"eqEsubset",
"gT",
"gacentEsd",
"group",
"im_invm",
"inE",
"inG",
"injG",
"injmSK",
"invm",
"invmK",
"mem_morphim",
"mem_morphpre",
"morphimJ",
"morphim_pHall",
"nGA'",
"on",
"pHall_sub",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ext_norm_conj_cent (H : {group gT}) x :
H \subset G -> x \in 'C_(G | to)(A) ->
[acts A, on H :^ x | to] = [acts A, on H | to]. | Proof.
move=> sHG /setIP[Gx].
rewrite gacentEsd !actsEsd ?conj_subG ?morphimJ // 2!inE Gx /=.
exact: norm_conj_cent.
Qed. | Lemma | ext_norm_conj_cent | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"actsEsd",
"conj_subG",
"gT",
"gacentEsd",
"group",
"inE",
"morphimJ",
"norm_conj_cent",
"on",
"sHG",
"setIP",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ext_coprime_Hall_subset (X : {group gT}) :
X \subset G -> pi.-group X -> [acts A, on X | to] ->
exists H : {group gT},
[/\ pi.-Hall(G) H, [acts A, on H | to] & X \subset H]. | Proof.
move=> sXG piX; rewrite actsEsd // => nXA'.
case: (coprime_Hall_subset nGA' coGA' solG' _ (morphim_pgroup _ piX) nXA').
exact: morphimS.
move=> H' /= [piH' nHA' sXH']; have sHG' := pHall_sub piH'.
exists (inG @*^-1 H')%G; rewrite actsEsd ?subsetIl ?morphpreK // nHA'.
rewrite -sub_morphim_pre //= sXH'; split=> ... | Lemma | ext_coprime_Hall_subset | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"Hall",
"actsEsd",
"coGA'",
"coprime_Hall_subset",
"gT",
"group",
"im_invm",
"inG",
"injG",
"morphimS",
"morphim_invmE",
"morphim_pHall",
"morphim_pgroup",
"morphpreK",
"nGA'",
"on",
"pHall_sub",
"pi",
"sXG",
"solG'",
"split",
"sub_morphim_pre",
"subsetIl",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ext_coprime_quotient_cent (H : {group gT}) :
H \subset G -> [acts A, on H | to] -> coprime #|H| #|A| -> solvable H ->
'C_(|to)(A) / H = 'C_(|to / H)(A). | Proof.
move=> sHG nHA coHA solH; pose N := 'N_G(H).
have nsHN: H <| N by rewrite normal_subnorm.
have [sHN nHn] := andP nsHN.
have sNG: N \subset G by apply: subsetIl.
have nNA: {acts A, on group N | to}.
split; rewrite // actsEsd // injm_subnorm ?injm_sdpair1 //=.
by rewrite normsI ?norms_norm ?im_sdpair_norm -?ac... | Lemma | ext_coprime_quotient_cent | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"A'",
"G'",
"actbyE",
"actsEsd",
"acts_act",
"acts_actby",
"apply",
"card_injm",
"coprime",
"coprime_TIg",
"coprime_norm_quotient_cent",
"coset",
"dom",
"domP",
"eqEsubset",
"gT",
"gacentEsd",
"gacentIdom",
"gacentIim",
"gacent_actby",
"gacent_ract",
"gact",
"group",
"i... | we do not require that G normalize H. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
sol_coprime_Sylow_exists A G :
solvable A -> A \subset 'N(G) -> coprime #|G| #|A| ->
exists2 P : {group gT}, p.-Sylow(G) P & A \subset 'N(P). | Proof.
move=> solA nGA coGA; pose AG := A <*> G.
have nsG_AG: G <| AG by rewrite /normal joing_subr join_subG nGA normG.
have [sG_AG nG_AG]:= andP nsG_AG.
have [P sylP] := Sylow_exists p G; pose N := 'N_AG(P); pose NG := G :&: N.
have nGN: N \subset 'N(G) by rewrite subIset ?nG_AG.
have sNG_G: NG \subset G := subsetIl ... | Lemma | sol_coprime_Sylow_exists | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"Frattini_arg",
"Hall",
"SchurZassenhaus_split",
"SchurZassenhaus_trans_actsol",
"Sylow",
"Sylow_exists",
"TI_cardMg",
"apply",
"card_quotient",
"coGA",
"complP",
"coprime",
"coprimeSg",
"coprime_cardMg",
"coprime_sym",
"divgI",
"divgS",
"gT",
"group",
"groupV",
"join_subG",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sol_coprime_Sylow_trans A G :
solvable A -> A \subset 'N(G) -> coprime #|G| #|A| ->
[transitive 'C_G(A), on [set P in 'Syl_p(G) | A \subset 'N(P)] | 'JG]. | Proof.
move=> solA nGA coGA; pose AG := A <*> G; set FpA := finset _.
have nG_AG: AG \subset 'N(G) by rewrite join_subG nGA normG.
have [P sylP nPA] := sol_coprime_Sylow_exists solA nGA coGA.
pose N := 'N_AG(P); have sAN: A \subset N by rewrite subsetI joing_subl.
have trNPA: A :^: AG ::&: N = A :^: N.
pose NG := 'N_... | Lemma | sol_coprime_Sylow_trans | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"SchurZassenhaus_trans_actsol",
"Syl_trans",
"actsP",
"afixJG",
"apply",
"astab1JG",
"atrans_supgroup",
"cardJg",
"cent_joinEr",
"coGA",
"conj_subG",
"coprime",
"coprimeSg",
"coprime_norm_cent",
"group_modl",
"imsetP",
"imset_f",
"inE",
"join_subG",
"joing_subl",
"joing_subr"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sol_coprime_Sylow_subset A G X :
A \subset 'N(G) -> coprime #|G| #|A| -> solvable A ->
X \subset G -> p.-group X -> A \subset 'N(X) ->
exists P : {group gT}, [/\ p.-Sylow(G) P, A \subset 'N(P) & X \subset P]. | Proof.
move=> nGA coGA solA sXG pX nXA.
pose nAp (Q : {group gT}) := [&& p.-group Q, Q \subset G & A \subset 'N(Q)].
have: nAp X by apply/and3P.
case/maxgroup_exists=> R; case/maxgroupP; case/and3P=> pR sRG nRA maxR sXR.
have [P sylP sRP]:= Sylow_superset sRG pR.
suffices defP: P :=: R by exists P; rewrite sylP defP.
c... | Lemma | sol_coprime_Sylow_subset | solvable | solvable/hall.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finset",
"prime",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"commutator",
"center",
"pgroup",
"finmodule",
"nilpotent",
"s... | [
"Hall_max",
"Sylow",
"Sylow_superset",
"apply",
"coGA",
"coprime",
"coprimeSg",
"defQ",
"gT",
"group",
"max_pgroup_Sylow",
"maxgroupP",
"maxgroup_exists",
"nRA",
"nilpotent_sub_norm",
"norm_sub_max_pgroup",
"normal_sub_max_pgroup",
"normal_subnorm",
"normsI",
"norms_norm",
"p... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
section (gT : finGroupType) | := GSection of {group gT} * {group gT}. | Inductive | section | solvable | solvable/jordanholder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gseries"
] | [
"gT",
"group"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mkSec (gT : finGroupType) (G1 G2 : {group gT}) | := GSection (G1, G2). | Definition | mkSec | solvable | solvable/jordanholder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gseries"
] | [
"G1",
"gT",
"group"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair_of_section gT (s : section gT) | := let: GSection u := s in u. | Coercion | pair_of_section | solvable | solvable/jordanholder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gseries"
] | [
"gT",
"section"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quotient_of_section gT (s : section gT) : GroupSet.sort _ | := s.1 / s.2. | Coercion | quotient_of_section | solvable | solvable/jordanholder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gseries"
] | [
"gT",
"section",
"sort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
section_group gT (s : section gT) : {group (coset_of s.2)} | :=
Eval hnf in [group of s]. | Coercion | section_group | solvable | solvable/jordanholder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gseries"
] | [
"coset_of",
"gT",
"group",
"section"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
section_group. | Canonical | section_group | solvable | solvable/jordanholder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gseries"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
section_isog | := [rel x y : section gT | x \isog y]. | Definition | section_isog | solvable | solvable/jordanholder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gseries"
] | [
"gT",
"isog",
"rel",
"section"
] | Isomorphic sections | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
section_repr s | := odflt (1 / 1)%sec (pick (section_isog ^~ s)). | Definition | section_repr | solvable | solvable/jordanholder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gseries"
] | [
"pick",
"section_isog"
] | A witness of the isomorphism class of a section | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
mksrepr G1 G2 | := section_repr (mkSec G1 G2). | Definition | mksrepr | solvable | solvable/jordanholder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gseries"
] | [
"G1",
"mkSec",
"section_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
section_reprP s : section_repr s \isog s. | Proof.
by rewrite /section_repr; case: pickP => //= /(_ s); rewrite isog_refl.
Qed. | Lemma | section_reprP | solvable | solvable/jordanholder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gseries"
] | [
"isog",
"isog_refl",
"pickP",
"section_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
section_repr_isog s1 s2 :
s1 \isog s2 -> section_repr s1 = section_repr s2. | Proof.
by move=> iso12; congr (odflt _ _); apply: eq_pick => s; apply: isog_transr.
Qed. | Lemma | section_repr_isog | solvable | solvable/jordanholder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gseries"
] | [
"apply",
"eq_pick",
"isog",
"isog_transr",
"s1",
"s2",
"section_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mkfactors (G : {group gT}) (s : seq {group gT}) | :=
map section_repr (pairmap (@mkSec _) G s). | Definition | mkfactors | solvable | solvable/jordanholder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gseries"
] | [
"gT",
"group",
"map",
"mkSec",
"pairmap",
"section_repr",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gTg | := {group gT}. | Notation | gTg | solvable | solvable/jordanholder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gseries"
] | [
"gT",
"group"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
compo | := [rel x y : {set gT} | maxnormal y x x]. | Notation | compo | solvable | solvable/jordanholder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gseries"
] | [
"gT",
"maxnormal",
"rel"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comps G s | := ((last G s) == 1%G) && compo.-series G s. | Definition | comps | solvable | solvable/jordanholder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gseries"
] | [
"compo",
"last"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
compsP G s :
reflect (last G s = 1%G /\ path [rel x y : gTg | maxnormal y x x] G s)
(comps G s). | Proof. by apply: (iffP andP) => [] [/eqP]. Qed. | Lemma | compsP | solvable | solvable/jordanholder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gseries"
] | [
"apply",
"comps",
"gTg",
"last",
"maxnormal",
"path",
"rel"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trivg_comps G s : comps G s -> (G :==: 1) = (s == [::]). | Proof.
case/andP=> ls cs; apply/eqP/eqP=> [G1 | s1]; last first.
by rewrite s1 /= in ls; apply/eqP.
by case: s {ls} cs => //= H s /andP[/maxgroupp]; rewrite G1 /proper sub1G andbF.
Qed. | Lemma | trivg_comps | solvable | solvable/jordanholder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gseries"
] | [
"G1",
"apply",
"comps",
"last",
"maxgroupp",
"proper",
"s1",
"sub1G"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comps_cons G H s : comps G (H :: s) -> comps H s. | Proof. by case/andP => /= ls /andP[_]; rewrite /comps ls. Qed. | Lemma | comps_cons | solvable | solvable/jordanholder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gseries"
] | [
"comps"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
simple_compsP G s : comps G s -> reflect (s = [:: 1%G]) (simple G). | Proof.
move=> cs; apply: (iffP idP) => [|s1]; last first.
by rewrite s1 /comps eqxx /= andbT -simple_maxnormal in cs.
case: s cs => [/trivg_comps/eqP-> | H s]; first by case/simpleP; rewrite eqxx.
rewrite [comps _ _]andbCA /= => /andP[/maxgroupp maxH /trivg_comps/esym nil_s].
rewrite simple_maxnormal => /maxgroupP[_ ... | Lemma | simple_compsP | solvable | solvable/jordanholder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gseries"
] | [
"apply",
"comps",
"eqxx",
"last",
"maxgroupP",
"maxgroupp",
"s1",
"simple",
"simpleP",
"simple_maxnormal",
"sub1G",
"trivg_comps",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exists_comps (G : gTg) : exists s, comps G s. | Proof.
elim: {G} #|G| {1 3}G (leqnn #|G|) => [G | n IHn G cG].
by rewrite leqNgt cardG_gt0.
have [sG | nsG] := boolP (simple G).
by exists [:: 1%G]; rewrite /comps eqxx /= -simple_maxnormal andbT.
have [-> | ntG] := eqVneq G 1%G; first by exists [::]; rewrite /comps eqxx.
have [N maxN] := ex_maxnormal_ntrivg ntG.
... | Lemma | exists_comps | solvable | solvable/jordanholder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gseries"
] | [
"apply",
"cardG_gt0",
"comps",
"eqVneq",
"eqxx",
"ex_maxnormal_ntrivg",
"gTg",
"leqNgt",
"leq_trans",
"leqnn",
"ltnS",
"maxnormal_proper",
"proper_card",
"sG",
"simple",
"simple_maxnormal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
JordanHolderUniqueness (G : gTg) (s1 s2 : seq gTg) :
comps G s1 -> comps G s2 -> perm_eq (mkfactors G s1) (mkfactors G s2). | Proof.
have [n] := ubnP #|G|; elim: n G => // n Hi G in s1 s2 * => /ltnSE-cG cs1 cs2.
have [G1 | ntG] := boolP (G :==: 1).
have -> : s1 = [::] by apply/eqP; rewrite -(trivg_comps cs1).
have -> : s2 = [::] by apply/eqP; rewrite -(trivg_comps cs2).
by rewrite /= perm_refl.
have [sG | nsG] := boolP (simple G).
by ... | Lemma | JordanHolderUniqueness | solvable | solvable/jordanholder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gseries"
] | [
"G1",
"apply",
"comps",
"eqxx",
"exists_comps",
"gTg",
"isog",
"isog_simple",
"isog_sym",
"leq_trans",
"ltnSE",
"maxnormalM",
"maxnormal_normal",
"maxnormal_proper",
"mkSec",
"mkfactors",
"mksrepr",
"nNG",
"normC",
"normal",
"normalS",
"normal_norm",
"normal_sub",
"norm... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gactsP (G : {set rT}) : reflect {acts A, on G | to} [acts A, on G | to]. | Proof.
apply: (iffP idP) => [nGA x|nGA]; first exact: acts_act.
apply/subsetP=> a Aa /[!inE]; rewrite Aa.
by apply/subsetP=> x; rewrite inE nGA.
Qed. | Lemma | gactsP | solvable | solvable/jordanholder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gseries"
] | [
"acts_act",
"apply",
"inE",
"on",
"subsetP",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gactsM (N1 N2 : {set rT}) :
N1 \subset D -> N2 \subset D ->
[acts A, on N1 | to] -> [acts A, on N2 | to] -> [acts A, on N1 * N2 | to]. | Proof.
move=> sN1D sN2D aAN1 aAN2; apply/gactsP=> x Ax y.
apply/idP/idP; case/mulsgP=> y1 y2 N1y1 N2y2 e.
move: (actKin to Ax y); rewrite e; move<-.
rewrite gactM ?groupV ?(subsetP sN1D y1) ?(subsetP sN2D) //.
by apply: mem_mulg; rewrite ?(gactsP _ aAN1) ?(gactsP _ aAN2) // groupV.
rewrite e gactM // ?(subsetP sN... | Lemma | gactsM | solvable | solvable/jordanholder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gseries"
] | [
"actKin",
"apply",
"gactM",
"gactsP",
"groupV",
"mem_mulg",
"mulsgP",
"on",
"subsetP",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gactsI (N1 N2 : {set rT}) :
[acts A, on N1 | to] -> [acts A, on N2 | to] -> [acts A, on N1 :&: N2 | to]. | Proof.
move=> aAN1 aAN2.
apply/subsetP=> x Ax; rewrite !inE Ax /=; apply/subsetP=> y Ny /[1!inE].
case/setIP: Ny=> N1y N2y; rewrite inE ?astabs_act ?N1y ?N2y //.
- by move/subsetP: aAN1; move/(_ x Ax).
- by move/subsetP: aAN2; move/(_ x Ax).
Qed. | Lemma | gactsI | solvable | solvable/jordanholder.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"gseries"
] | [
"apply",
"astabs_act",
"inE",
"on",
"setIP",
"subsetP",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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