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Fitting_pcore pi G : 'F('O_pi(G)) = 'O_pi('F(G)).
Proof. apply/eqP; rewrite eqEsubset. rewrite (subset_trans _ (pcoreS _ (Fitting_sub _))). by rewrite subsetI Fitting_sub Fitting_max ?Fitting_nil ?gFnormal_trans. rewrite (subset_trans _ (FittingS (pcore_sub _ _))) // subsetI pcore_sub. by rewrite pcore_max ?pcore_pgroup ?gFnormal_trans. Qed.
Lemma
Fitting_pcore
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "FittingS", "Fitting_max", "Fitting_nil", "Fitting_sub", "apply", "eqEsubset", "gFnormal_trans", "pcoreS", "pcore_max", "pcore_pgroup", "pcore_sub", "pi", "subsetI", "subset_trans" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
index_maxnormal_sol_prime (H : {group gT}) : solvable G -> maxnormal H G G -> prime #|G : H|.
Proof. move=> solG maxH; have nsHG := maxnormal_normal maxH. rewrite -card_quotient ?normal_norm // simple_sol_prime ?quotient_sol //. by rewrite quotient_simple. Qed.
Lemma
index_maxnormal_sol_prime
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "card_quotient", "gT", "group", "maxnormal", "maxnormal_normal", "normal_norm", "nsHG", "prime", "quotient_simple", "quotient_sol", "simple_sol_prime", "solvable" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sol_prime_factor_exists : solvable G -> G :!=: 1 -> {H : {group gT} | H <| G & prime #|G : H| }.
Proof. move=> solG /ex_maxnormal_ntrivg[H maxH]. by exists H; [apply: maxnormal_normal | apply: index_maxnormal_sol_prime]. Qed.
Lemma
sol_prime_factor_exists
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "apply", "ex_maxnormal_ntrivg", "gT", "group", "index_maxnormal_sol_prime", "maxnormal_normal", "prime", "solvable" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
center_special_abelem : p.-group G -> special G -> p.-abelem 'Z(G).
Proof. move=> pG [defPhi defG']. have [-> | ntG] := eqsVneq G 1; first by rewrite center1 abelem1. have [p_pr _ _] := pgroup_pdiv pG ntG. have fM: {in 'Z(G) &, {morph natexp^~ p : x y / x * y}}. by move=> x y /setIP[_ /centP cxG] /setIP[/cxG cxy _]; apply: expgMn. rewrite abelemE //= center_abelian; apply/exponentP=>...
Lemma
center_special_abelem
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "MhoE", "Phi_joing", "abelem", "abelem1", "abelemE", "apply", "centP", "center1", "center_abelian", "commXg", "commgP", "eqsVneq", "expgMn", "expn1", "exponentP", "fM", "gen_subG", "group", "imset2P", "imset_f", "inE", "joing_idr", "kerP", "mem_commg", "mem_gen", "m...
This is Aschbacher (23.7)
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
exponent_special : p.-group G -> special G -> exponent G %| p ^ 2.
Proof. move=> pG spG; have [defPhi _] := spG. have /and3P[_ _ expZ] := center_special_abelem pG spG. apply/exponentP=> x Gx; rewrite expgM (exponentP expZ) // -defPhi. by rewrite (Phi_joing pG) mem_gen // inE orbC (Mho_p_elt 1) ?(mem_p_elt pG). Qed.
Lemma
exponent_special
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "Mho_p_elt", "Phi_joing", "apply", "center_special_abelem", "expgM", "exponent", "exponentP", "group", "inE", "mem_gen", "mem_p_elt", "pG", "special" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelian_charsimple_special : p.-group G -> coprime #|G| #|A| -> [~: G, A] = G -> \bigcup_(H : {group gT} | (H \char G) && abelian H) H \subset 'C(A) -> special G /\ 'C_G(A) = 'Z(G).
Proof. move=> pG coGA defG /bigcupsP cChaA. have cZA: 'Z(G) \subset 'C_G(A). by rewrite subsetI center_sub cChaA // center_char center_abelian. have cChaG (H : {group gT}): H \char G -> abelian H -> H \subset 'Z(G). move=> chH abH; rewrite subsetI char_sub //= centsC -defG. rewrite comm_norm_cent_cent ?(char_norm...
Theorem
abelian_charsimple_special
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "MhoE", "Mho_char", "Mho_p_elt", "Mho_sub", "Phi_joing", "TI_center_nil", "abelian", "add1n", "addSn", "addn1", "apply", "bigcupsP", "centS", "cent_gen", "center_abelian", "center_char", "center_sub", "centsC", "centsP", "char", "charI", "char_norm", "char_sub", "coGA",...
Aschbacher 24.7 (replaces Gorenstein 5.3.7)
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
pZ : p.-group 'Z(S)
:= pgroupS (center_sub S) pS.
Let
pZ
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "center_sub", "group", "pS", "pgroupS" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
extraspecial_prime : prime p.
Proof. by case: esS => _ /prime_gt1; rewrite cardG_gt1; case/(pgroup_pdiv pZ). Qed.
Lemma
extraspecial_prime
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "cardG_gt1", "pZ", "pgroup_pdiv", "prime", "prime_gt1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_center_extraspecial : #|'Z(S)| = p.
Proof. by apply/eqP; apply: (pgroupP pZ); case: esS. Qed.
Lemma
card_center_extraspecial
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "apply", "pZ", "pgroupP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
min_card_extraspecial : #|S| >= p ^ 3.
Proof. have p_gt1 := prime_gt1 extraspecial_prime. rewrite leqNgt (card_pgroup pS) ltn_exp2l // ltnS. case: esS => [[_ defS']]; apply: contraL => /(p2group_abelian pS)/derG1P S'1. by rewrite -defS' S'1 cards1. Qed.
Lemma
min_card_extraspecial
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "apply", "card_pgroup", "cards1", "derG1P", "extraspecial_prime", "leqNgt", "ltnS", "ltn_exp2l", "p2group_abelian", "pS", "p_gt1", "prime_gt1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_p3group_extraspecial E : prime p -> #|E| = (p ^ 3)%N -> #|'Z(E)| = p -> extraspecial E.
Proof. move=> p_pr oEp3 oZp; have p_gt0 := prime_gt0 p_pr. have pE: p.-group E by rewrite /pgroup oEp3 pnatX pnat_id. have pEq: p.-group (E / 'Z(E))%g by rewrite quotient_pgroup. have /andP[sZE nZE] := center_normal E. have oEq: #|E / 'Z(E)|%g = (p ^ 2)%N. by rewrite card_quotient -?divgS // oEp3 oZp expnS mulKn. hav...
Lemma
card_p3group_extraspecial
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "Phi_joing", "TI_center_nil", "abelemP", "abelian", "apply", "card_p2group_abelian", "card_pgroup", "card_quotient", "center_normal", "commG1P", "cycle_subG", "cyclicP", "cyclic_center_factor_abelian", "der1_min", "der_normal", "divgS", "dvdn_exp2l", "dvdn_leq", "eqEcard", "eqE...
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
p3group_extraspecial G : p.-group G -> ~~ abelian G -> logn p #|G| <= 3 -> extraspecial G.
Proof. move=> pG not_cGG; have /andP[sZG nZG] := center_normal G. have ntG: G :!=: 1 by apply: contraNneq not_cGG => ->; apply: abelian1. have ntZ: 'Z(G) != 1 by rewrite (center_nil_eq1 (pgroup_nil pG)). have [p_pr _ [n oG]] := pgroup_pdiv pG ntG; rewrite oG pfactorK //. have [_ _ [m oZ]] := pgroup_pdiv (pgroupS sZG pG...
Lemma
p3group_extraspecial
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "abelian", "abelian1", "addSn", "addn1", "apply", "cardSg", "card_p3group_extraspecial", "card_pgroup", "card_quotient", "center_nil_eq1", "center_normal", "contraNneq", "cyclic_center_factor_abelian", "divgS", "dvdn_exp2l", "dvdn_prime_cyclic", "extraspecial", "group", "leqn0", ...
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
extraspecial_nonabelian G : extraspecial G -> ~~ abelian G.
Proof. case=> [[_ defG'] oZ]; rewrite /abelian (sameP commG1P eqP). by rewrite -derg1 defG' -cardG_gt1 prime_gt1. Qed.
Lemma
extraspecial_nonabelian
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "abelian", "cardG_gt1", "commG1P", "derg1", "extraspecial", "oZ", "prime_gt1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
exponent_2extraspecial G : 2.-group G -> extraspecial G -> exponent G = 4.
Proof. move=> p2G esG; have [spG _] := esG. case/dvdn_pfactor: (exponent_special p2G spG) => // k. rewrite leq_eqVlt ltnS => /predU1P[-> // | lek1] expG. case/negP: (extraspecial_nonabelian esG). by rewrite (@abelem_abelian _ 2) ?exponent2_abelem // expG pfactor_dvdn. Qed.
Lemma
exponent_2extraspecial
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "abelem_abelian", "dvdn_pfactor", "exponent", "exponent2_abelem", "exponent_special", "extraspecial", "extraspecial_nonabelian", "group", "leq_eqVlt", "ltnS", "pfactor_dvdn", "predU1P" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_special D G (f : {morphism D >-> rT}) : 'injm f -> G \subset D -> special G -> special (f @* G).
Proof. move=> injf sGD [defPhiG defG']. by rewrite /special -morphim_der // -injm_Phi // defPhiG defG' injm_center. Qed.
Lemma
injm_special
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "injf", "injm_Phi", "injm_center", "morphim_der", "morphism", "sGD", "special" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_extraspecial D G (f : {morphism D >-> rT}) : 'injm f -> G \subset D -> extraspecial G -> extraspecial (f @* G).
Proof. move=> injf sGD [spG ZG_pr]; split; first exact: injm_special spG. by rewrite -injm_center // card_injm // subIset ?sGD. Qed.
Lemma
injm_extraspecial
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "card_injm", "extraspecial", "injf", "injm_center", "injm_special", "morphism", "sGD", "split", "subIset" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog_special G (R : {group rT}) : G \isog R -> special G -> special R.
Proof. by case/isogP=> f injf <-; apply: injm_special. Qed.
Lemma
isog_special
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "apply", "group", "injf", "injm_special", "isog", "isogP", "special" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog_extraspecial G (R : {group rT}) : G \isog R -> extraspecial G -> extraspecial R.
Proof. by case/isogP=> f injf <-; apply: injm_extraspecial. Qed.
Lemma
isog_extraspecial
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "apply", "extraspecial", "group", "injf", "injm_extraspecial", "isog", "isogP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
cprod_extraspecial G H K : p.-group G -> H \* K = G -> H :&: K = 'Z(H) -> extraspecial H -> extraspecial K -> extraspecial G.
Proof. move=> pG defG ziHK [[PhiH defH'] ZH_pr] [[PhiK defK'] ZK_pr]. have [_ defHK cHK]:= cprodP defG. have sZHK: 'Z(H) \subset 'Z(K). by rewrite subsetI -{1}ziHK subsetIr subIset // centsC cHK. have{sZHK} defZH: 'Z(H) = 'Z(K). by apply/eqP; rewrite eqEcard sZHK leq_eqVlt eq_sym -dvdn_prime2 ?cardSg. have defZ: 'Z...
Lemma
cprod_extraspecial
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "Phi_cprod", "apply", "cardSg", "center_cprod", "centsC", "cprodP", "defG", "der_cprod", "dvdn_prime2", "eqEcard", "eq_sym", "extraspecial", "group", "leq_eqVlt", "mulGid", "pG", "split", "subIset", "subsetI", "subsetIr" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
(pG : p.-group G) (esG : extraspecial G).
Hypotheses
pG
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "extraspecial", "group" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_pr
:= extraspecial_prime pG esG.
Let
p_pr
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "extraspecial_prime", "pG" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
oZ
:= card_center_extraspecial pG esG.
Let
oZ
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "card_center_extraspecial", "pG" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
cent1_extraspecial_maximal x : x \in G -> x \notin 'Z(G) -> maximal 'C_G[x] G.
Proof. move=> Gx notZx; pose f y := [~ x, y]; have [[_ defG'] prZ] := esG. have{defG'} fZ y: y \in G -> f y \in 'Z(G). by move=> Gy; rewrite -defG' mem_commg. have fM: {in G &, {morph f : y z / y * z}}%g. move=> y z Gy Gz; rewrite {1}/f commgMJ conjgCV -conjgM (conjg_fixP _) //. rewrite (sameP commgP cent1P); app...
Lemma
cent1_extraspecial_maximal
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "apply", "card_isog", "card_quotient", "cent1P", "centP", "centS", "cent_set1", "commgMJ", "commgP", "conjgCV", "conjgM", "conjg_fixP", "eqEsubset", "eq_invg_mul", "fM", "first_isog", "groupM", "groupV", "inE", "in_set1", "invg_comm", "ker", "ker_norm", "maximal", "me...
This encasulates Aschbacher (23.10)(1).
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
subcent1_extraspecial_maximal U x : U \subset G -> x \in G :\: 'C(U) -> maximal 'C_U[x] U.
Proof. move=> sUG /setDP[Gx not_cUx]; apply/maxgroupP; split=> [|H ltHU sCxH]. by rewrite /proper subsetIl subsetI subxx sub_cent1. case/andP: ltHU => sHU not_sHU; have sHG := subset_trans sHU sUG. apply/eqP; rewrite eqEsubset sCxH subsetI sHU /= andbT. apply: contraR not_sHU => not_sHCx. have maxCx: maximal 'C_G[x] ...
Lemma
subcent1_extraspecial_maximal
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "apply", "cent1_extraspecial_maximal", "centS", "eqEsubset", "group_modr", "inE", "maxgroupP", "maximal", "mul_subG", "mulg_normal_maximal", "pG", "p_maximal_normal", "proper", "sHG", "setDP", "setIA", "setIidPl", "split", "sub_cent1", "subsetI", "subsetIl", "subsetP", "s...
(19.1)) to subgroups of an extraspecial group.
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_subcent_extraspecial U : U \subset G -> #|'C_G(U)| = (#|'Z(G) :&: U| * #|G : U|)%N.
Proof. move=> sUG; rewrite setIAC (setIidPr sUG). have [m leUm] := ubnP #|U|; elim: m => // m IHm in U leUm sUG *. have [cUG | not_cUG]:= orP (orbN (G \subset 'C(U))). by rewrite !(setIidPl _) ?Lagrange // centsC. have{not_cUG} [x Gx not_cUx] := subsetPn not_cUG. pose W := 'C_U[x]; have sCW_G: 'C_G(W) \subset G := su...
Lemma
card_subcent_extraspecial
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "Lagrange", "Lagrange_index", "Uu", "apply", "cent1C", "centM", "centS", "cent_cycle", "cent_set1", "centsC", "cycle_subG", "defU", "eqn_pmul2r", "inE", "leq_trans", "maxgroupp", "maximal", "mulg_normal_maximal", "mulnA", "pG", "p_gt0", "p_maximal_index", "p_maximal_norma...
(Aschbacher (19.2)) to subgroups of an extraspecial group.
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
split1_extraspecial x : x \in G :\: 'Z(G) -> {E : {group gT} & {R : {group gT} | [/\ #|E| = (p ^ 3)%N /\ #|R| = #|G| %/ p ^ 2, E \* R = G /\ E :&: R = 'Z(E), 'Z(E) = 'Z(G) /\ 'Z(R) = 'Z(G), extraspecial E /\ x \in E & if abelian R then R :=: 'Z(G) else extraspecial R]}}.
Proof. case/setDP=> Gx notZx; rewrite inE Gx /= in notZx. have [[defPhiG defG'] prZ] := esG. have maxCx: maximal 'C_G[x] G. by rewrite subcent1_extraspecial_maximal // inE notZx. pose y := repr (G :\: 'C[x]). have [Gy not_cxy]: y \in G /\ y \notin 'C[x]. move/maxgroupp: maxCx => /properP[_ [t Gt not_cyt]]. by app...
Lemma
split1_extraspecial
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "PhiS", "Phi_joing", "abelian", "abelianE", "apply", "cardG_gt0", "card_p3group_extraspecial", "cent1C", "cent1P", "cent1id", "centM", "centS", "centY", "cent_cycle", "cent_joinEr", "center_idP", "center_prod", "center_sub", "commG1P", "commgP", "cprodE", "cycle_id", "cyc...
group.
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
pmaxElem_extraspecial : 'E*_p(G) = 'E_p^('r_p(G))(G).
Proof. have sZmax: {in 'E*_p(G), forall E, 'Z(G) \subset E}. move=> E maxE; have defE := pmaxElem_LdivP p_pr maxE. have abelZ: p.-abelem 'Z(G) by rewrite prime_abelem ?oZ. rewrite -(Ohm1_id abelZ) (OhmE 1 (abelem_pgroup abelZ)) gen_subG -defE. by rewrite setSI // setIS ?centS // -defE !subIset ?subxx. suffices ...
Lemma
pmaxElem_extraspecial
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "Lagrange", "Ohm1_id", "OhmE", "TI_cardMg", "abelE", "abelem", "abelemS", "abelem_pgroup", "abelem_split_dprod", "apply", "cardG_gt0", "card_subcent_extraspecial", "centM", "centS", "dprodP", "eqEsubset", "eqn_leq", "gen_subG", "inE", "last", "leq_mul", "leq_pmul2l", "leq...
Note that Aschbacher derives this from the Witt lemma, which we avoid.
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
critical_extraspecial R S : p.-group R -> S \subset R -> extraspecial S -> [~: S, R] \subset S^`(1) -> S \* 'C_R(S) = R.
Proof. move=> pR sSR esS sSR_S'; have [[defPhi defS'] _] := esS. have [pS [sPS nPS]] := (pgroupS sSR pR, andP (Phi_normal S : 'Phi(S) <| S)). have{esS} oZS: #|'Z(S)| = p := card_center_extraspecial pS esS. have nSR: R \subset 'N(S) by rewrite -commg_subl (subset_trans sSR_S') ?der_sub. have nsCR: 'C_R(S) <| R by rewrit...
Lemma
critical_extraspecial
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "Aut_aut", "Phi_nongen", "Phi_normal", "Phi_quotient_abelem", "apply", "cardG_gt0", "card_center_extraspecial", "card_in_imset", "card_isog", "card_pffun_on", "card_pgroup", "cent1C", "centP", "cent_gen", "cent_normal", "centsC", "cents_norm", "commg_subl", "conjg", "conjgK", ...
This is B & G, Theorem 4.15, as done in Aschbacher (23.8)
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
extraspecial_structure S : p.-group S -> extraspecial S -> {Es | all (fun E => (#|E| == p ^ 3)%N && ('Z(E) == 'Z(S))) Es & \big[cprod/1%g]_(E <- Es) E \* 'Z(S) = S}.
Proof. have [m] := ubnP #|S|; elim: m S => // m IHm S leSm pS esS. have [x Z'x]: {x | x \in S :\: 'Z(S)}. apply/sigW/set0Pn; rewrite -subset0 subDset setU0. apply: contra (extraspecial_nonabelian esS) => sSZ. exact: abelianS sSZ (center_abelian S). have [E [R [[oE oR]]]]:= split1_extraspecial pS esS Z'x. case=> d...
Theorem
extraspecial_structure
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "abelianS", "all", "apply", "big_cons", "big_seq1", "cardG_gt0", "center_abelian", "cprod", "cprodA", "cprodP", "defR", "eqxx", "extraspecial", "extraspecial_nonabelian", "extraspecial_prime", "group", "leq_trans", "ltn_Pdiv", "ltn_exp2l", "mulG_subr", "pS", "pgroupS", "p...
This is part of Aschbacher (23.13) and (23.14).
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
oZ
:= card_center_extraspecial pS esS.
Let
oZ
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "card_center_extraspecial", "pS" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_extraspecial : {n | n > 0 & #|S| = (p ^ n.*2.+1)%N}.
Proof. set T := S; exists (logn p #|T|)./2. rewrite half_gt0 ltnW // -(leq_exp2l _ _ (prime_gt1 p_pr)) -card_pgroup //. exact: min_card_extraspecial. have [Es] := extraspecial_structure pS esS; rewrite -[in RHS]/T. elim: Es T => [_ _ <-| E s IHs T] /=. by rewrite big_nil cprod1g oZ (pfactorK 1). rewrite -andbA bi...
Lemma
card_extraspecial
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "addSn", "apply", "big_cons", "big_nil", "cardG_gt0", "card_pgroup", "cprod1g", "cprodA", "cprodP", "defU", "eqEsubset", "expnD", "expnS", "extraspecial_structure", "half_gt0", "leq_exp2l", "logn", "ltnW", "min_card_extraspecial", "mulG_subr", "mulKn", "mul_cardG", "mulnK...
This is Aschbacher (23.10)(2).
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Aut_extraspecial_full : Aut_in (Aut S) 'Z(S) \isog Aut 'Z(S).
Proof. have [p_gt1 p_gt0] := (prime_gt1 p_pr, prime_gt0 p_pr). have [Es] := extraspecial_structure pS esS. elim: Es S oZ => [T _ _ <-| E s IHs T oZT] /=. rewrite big_nil cprod1g (center_idP (center_abelian T)). by apply/Aut_sub_fullP=> // g injg gZ; exists g. rewrite -andbA big_cons -cprodA => /and3P[/eqP-oE /eqP-d...
Lemma
Aut_extraspecial_full
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "Aut", "Aut_aut", "Aut_cprod_full", "Aut_in", "Aut_in_isog", "Aut_sub_fullP", "Mho_p_elt", "Phi_joing", "abelian", "abelianS", "apply", "aut", "autE", "big_cons", "big_nil", "bin2odd", "cardG_gt0", "card_Aut_cyclic", "card_im_injm", "card_p2group_abelian", "card_p3group_extra...
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
center_aut_extraspecial k : coprime k p -> exists2 f, f \in Aut S & forall z, z \in 'Z(S) -> f z = (z ^+ k)%g.
Proof. have /cyclicP[z defZ]: cyclic 'Z(S) by rewrite prime_cyclic ?oZ. have oz: #[z] = p by rewrite orderE -defZ. rewrite coprime_sym -unitZpE ?prime_gt1 // -oz => u_k. pose g := Zp_unitm (FinRing.unit 'Z_#[z] u_k). have AutZg: g \in Aut 'Z(S) by rewrite defZ -im_Zp_unitm mem_morphim ?inE. have ZSfull := Aut_sub_fullP...
Lemma
center_aut_extraspecial
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "Aut", "Aut_aut", "Aut_extraspecial_full", "Aut_sub_fullP", "Zp_unitm", "aut", "autE", "autmE", "center_sub", "coprime", "coprime_sym", "cyclem", "cyclic", "cyclicP", "expg_znat", "im_Zp_unitm", "im_autm", "inE", "injf", "injm_autm", "mem_morphim", "oZ", "orderE", "perm...
quoted from Gorenstein in the proof of B & G, Theorem 2.5.
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
SCN_P A : reflect (A <| G /\ 'C_G(A) = A) (A \in 'SCN(G)).
Proof. by apply: (iffP setIdP) => [] [->]; move/eqP. Qed.
Lemma
SCN_P
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "SCN", "apply", "setIdP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
SCN_abelian A : A \in 'SCN(G) -> abelian A.
Proof. by case/SCN_P=> _ defA; rewrite /abelian -{1}defA subsetIr. Qed.
Lemma
SCN_abelian
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "SCN", "SCN_P", "abelian", "subsetIr" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
exponent_Ohm1_class2 H : odd p -> p.-group H -> nil_class H <= 2 -> exponent 'Ohm_1(H) %| p.
Proof. move=> odd_p pH; rewrite nil_class2 => sH'Z; apply/exponentP=> x /=. rewrite (OhmE 1 pH) expn1 gen_set_id => {x} [|/LdivP[] //]. apply/group_setP; split=> [|x y]; first by rewrite !inE group1 expg1n //=. case/LdivP=> Hx xp1 /LdivP[Hy yp1]; rewrite !inE groupM //=. have [_ czH]: [~ y, x] \in H /\ centralises [~ y...
Lemma
exponent_Ohm1_class2
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "LdivP", "OhmE", "apply", "bin2odd", "centerP", "centralises", "comm1g", "commXXg", "commute", "expMg_Rmul", "expg1n", "expn1", "exponent", "exponentP", "gen_set_id", "group", "group1", "groupM", "group_setP", "inE", "mem_commg", "mul1g", "nil_class", "nil_class2", "o...
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
SCN_max A : A \in 'SCN(G) -> [max A | A <| G & abelian A].
Proof. case/SCN_P => nAG scA; apply/maxgroupP; split=> [|H]. by rewrite nAG /abelian -{1}scA subsetIr. do 2![case/andP] => sHG _ abelH sAH; apply/eqP. by rewrite eqEsubset sAH -scA subsetI sHG centsC (subset_trans sAH). Qed.
Lemma
SCN_max
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "SCN", "SCN_P", "abelH", "abelian", "apply", "centsC", "eqEsubset", "max", "maxgroupP", "sHG", "split", "subsetI", "subsetIr", "subset_trans" ]
SCN_max and max_SCN cover Aschbacher 23.15(1)
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
max_SCN A : p.-group G -> [max A | A <| G & abelian A] -> A \in 'SCN(G).
Proof. move/pgroup_nil=> nilG; rewrite /abelian. case/maxgroupP=> /andP[nsAG abelA] maxA; have [sAG nAG] := andP nsAG. rewrite inE nsAG eqEsubset /= andbC subsetI abelA normal_sub //=. rewrite -quotient_sub1; first by rewrite subIset 1?normal_norm. apply/trivgP; apply: (TI_center_nil (quotient_nil A nilG)). by rewrit...
Lemma
max_SCN
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "SCN", "TI_center_nil", "abelian", "apply", "center", "center_prod", "centsC", "cents_norm", "coset", "cycle_abelian", "cycle_subG", "eqEsubset", "group", "inE", "inv_quotientN", "max", "maxA", "maxgroupP", "morphimP", "normG", "normal", "normalS", "normal_norm", "norma...
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
SCN_A : A \in 'SCN(G).
Hypothesis
SCN_A
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "SCN" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Z
:= 'Ohm_1(A).
Let
Z
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
cAA
:= SCN_abelian SCN_A.
Let
cAA
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "SCN_A", "SCN_abelian" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sZA: Z \subset A
:= Ohm_sub 1 A.
Let
sZA
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "Ohm_sub" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
nZA : A \subset 'N(Z)
:= sub_abelian_norm cAA sZA.
Let
nZA
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "cAA", "sZA", "sub_abelian_norm" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
der1_stab_Ohm1_SCN_series : ('C(Z) :&: 'C_G(A / Z | 'Q))^`(1) \subset A.
Proof. case/SCN_P: SCN_A => /andP[sAG nAG] {4} <-. rewrite subsetI {1}setICA comm_subG ?subsetIl //= gen_subG. apply/subsetP=> w /imset2P[u v]. rewrite /= -groupV -(groupV _ v) /= astabQR //= -/Z !inE (groupV 'C(Z)). case/and4P=> cZu _ _ sRuZ /and4P[cZv' _ _ sRvZ] ->{w}. apply/centP=> a Aa; rewrite /commute -!mulgA (co...
Lemma
der1_stab_Ohm1_SCN_series
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "SCN_A", "SCN_P", "apply", "astabQR", "centP", "comm_subG", "commgC", "commgCV", "commute", "gen_subG", "groupV", "imset2P", "inE", "invgK", "mem_commg", "mulKVg", "mulKg", "mulgA", "sAG", "set11", "setICA", "subsetI", "subsetIl", "subsetP" ]
This is Aschbacher 23.15(2).
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ohm1_stab_Ohm1_SCN_series : odd p -> p.-group G -> 'Ohm_1('C_G(Z)) \subset 'C_G(A / Z | 'Q).
Proof. have [-> | ntG] := eqsVneq G 1; first by rewrite !(setIidPl (sub1G _)) Ohm1. move=> p_odd pG; have{ntG} [p_pr _ _] := pgroup_pdiv pG ntG. case/SCN_P: SCN_A => /andP[sAG nAG] _; have pA := pgroupS sAG pG. have pCGZ : p.-group 'C_G(Z) by rewrite (pgroupS _ pG) // subsetIl. rewrite {pCGZ}(OhmE 1 pCGZ) gen_subG; app...
Lemma
Ohm1_stab_Ohm1_SCN_series
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "Ohm1", "OhmE", "OhmS", "Ohm_id", "Ohm_sub", "SCN_A", "SCN_P", "abelianM", "abelianS", "apply", "astabQ", "cAA", "cent_norm", "cent_sub", "centsC", "cents_norm", "coprime_TIg", "cycle_abelian", "cycle_subG", "der1_min", "dvdn_quotient", "dvdn_trans", "eqEsubset", "eqsVn...
maximality of A.
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ohm1_cent_max_normal_abelem Z : odd p -> p.-group G -> [max Z | Z <| G & p.-abelem Z] -> 'Ohm_1('C_G(Z)) = Z.
Proof. move=> p_odd pG; set X := 'Ohm_1('C_G(Z)). case/maxgroupP=> /andP[nsZG abelZ] maxZ. have [sZG nZG] := andP nsZG; have [_ cZZ expZp] := and3P abelZ. have{nZG} nsXG: X <| G by rewrite gFnormal_trans ?norm_normalI ?norms_cent. have cZX : X \subset 'C(Z) by apply/gFsub_trans/subsetIr. have{sZG expZp} sZX: Z \subset ...
Lemma
Ohm1_cent_max_normal_abelem
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "Ohm1_abelem", "Ohm1_id", "Ohm1_stab_Ohm1_SCN_series", "OhmE", "OhmS", "Ohm_id", "Ohm_sub", "SCN", "SCN_A", "TI_center_nil", "abelem", "abelian", "apply", "bigcupsP", "cAA", "cardJg", "centS", "centsC", "cents_norm", "class_support", "class_supportEr", "commgSS", "conjg1"...
This is Aschbacher 23.16.
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
critical_class2 H : critical H G -> nil_class H <= 2.
Proof. case=> [chH _ sRZ _]. by rewrite nil_class2 (subset_trans _ sRZ) ?commSg // char_sub. Qed.
Lemma
critical_class2
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "char_sub", "commSg", "critical", "nil_class", "nil_class2", "subset_trans" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Thompson_critical : p.-group G -> {K : {group gT} | critical K G}.
Proof. move=> pG; pose qcr A := (A \char G) && ('Phi(A) :|: [~: G, A] \subset 'Z(A)). have [|K]:= @maxgroup_exists _ qcr 1 _. by rewrite /qcr char1 center1 commG1 subUset Phi_sub subxx. case/maxgroupP; rewrite {}/qcr subUset => /and3P[chK sPhiZ sRZ] maxK _. have sKG := char_sub chK; have nKG := char_normal chK. exist...
Lemma
Thompson_critical
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "Ohm1_abelem", "Phi_mulg", "Phi_sub", "TI_Ohm1", "TI_center_nil", "abelemS", "apply", "cent_joinEr", "center1", "center_abelian", "center_prod", "center_sub", "centsC", "cents_norm", "char", "char1", "charI", "charM", "char_from_quotient", "char_norm", "char_normal", "char_...
This proof of the Thompson critical lemma is adapted from Aschbacher 23.6
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
critical_p_stab_Aut H : critical H G -> p.-group G -> p.-group 'C(H | [Aut G]).
Proof. move=> [chH sPhiZ sRZ eqCZ] pG; have sHG := char_sub chH. pose G' := (sdpair1 [Aut G] @* G)%G; pose H' := (sdpair1 [Aut G] @* H)%G. apply/pgroupP=> q pr_q; case/Cauchy=> //= f cHF; move: (cHF); rewrite astab_ract. case/setIP=> Af cHFP ofq; rewrite -cycle_subG in cHF; apply: (pgroupP pG) => //. pose F' := (sdpair...
Lemma
critical_p_stab_Aut
solvable
solvable/maximal.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "commutator", "gproduct", "gfunctor", "ssralg",...
[ "Aut", "Cauchy", "G'", "apply", "astabP", "astab_act", "astab_ract", "autmE", "center_sub", "char_norm", "char_sub", "comm1G", "commG1P", "commGC", "commSg", "commgEl", "commg_subr", "critical", "cycle_id", "cycle_subG", "dvd1n", "eq_mulVg1", "expgS", "expgSr", "expg_...
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lower_central_at
:= iter n.-1 (fun B => [~: B, A]) A.
Definition
lower_central_at
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "iter" ]
starts at 0 (sic).
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
upper_central_at
:= iter n (fun B => coset B @*^-1 'Z(A / B)) 1.
Definition
upper_central_at
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "coset", "iter" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''L_' n ( G )"
:= (lower_central_at n G) (n at level 2, format "''L_' n ( G )") : group_scope.
Notation
''L_' n ( G )
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "lower_central_at" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''Z_' n ( G )"
:= (upper_central_at n G) (n at level 2, format "''Z_' n ( G )") : group_scope.
Notation
''Z_' n ( G )
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "upper_central_at" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
nilpotent
:= [forall (G : {group gT} | G \subset A :&: [~: G, A]), G :==: 1].
Definition
nilpotent
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "gT", "group" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
nil_class
:= index 1 (mkseq (fun n => 'L_n.+1(A)) #|A|).
Definition
nil_class
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "index", "mkseq" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
solvable
:= [forall (G : {group gT} | G \subset A :&: [~: G, G]), G :==: 1].
Definition
solvable
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "gT", "group" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
nilpotent1 : nilpotent [1 gT].
Proof. by apply/forall_inP=> H; rewrite commG1 setIid -subG1. Qed.
Lemma
nilpotent1
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "apply", "commG1", "forall_inP", "gT", "nilpotent", "setIid", "subG1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
nilpotentS A B : B \subset A -> nilpotent A -> nilpotent B.
Proof. move=> sBA nilA; apply/forall_inP=> H sHR. have:= forallP nilA H; rewrite (subset_trans sHR) //. by apply: subset_trans (setIS _ _) (setSI _ _); rewrite ?commgS. Qed.
Lemma
nilpotentS
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "apply", "commgS", "forallP", "forall_inP", "nilpotent", "setIS", "setSI", "subset_trans" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
nil_comm_properl G H A : nilpotent G -> H \subset G -> H :!=: 1 -> A \subset 'N_G(H) -> [~: H, A] \proper H.
Proof. move=> nilG sHG ntH; rewrite subsetI properE; case/andP=> sAG nHA. rewrite (subset_trans (commgS H (subset_gen A))) ?commg_subl ?gen_subG //. apply: contra ntH => sHR; have:= forallP nilG H; rewrite subsetI sHG. by rewrite (subset_trans sHR) ?commgS. Qed.
Lemma
nil_comm_properl
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "apply", "commgS", "commg_subl", "forallP", "gen_subG", "nilpotent", "proper", "properE", "sAG", "sHG", "subsetI", "subset_gen", "subset_trans" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
nil_comm_properr G A H : nilpotent G -> H \subset G -> H :!=: 1 -> A \subset 'N_G(H) -> [~: A, H] \proper H.
Proof. by rewrite commGC; apply: nil_comm_properl. Qed.
Lemma
nil_comm_properr
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "apply", "commGC", "nil_comm_properl", "nilpotent", "proper" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
centrals_nil (s : seq {group gT}) G : G.-central.-series 1%G s -> last 1%G s = G -> nilpotent G.
Proof. move=> cGs defG; apply/forall_inP=> H /subsetIP[sHG sHR]. move: sHG; rewrite -{}defG -subG1 -[1]/(gval 1%G). elim: s 1%G cGs => //= L s IHs K /andP[/and3P[sRK sKL sLG] /IHs sHL] sHs. exact: subset_trans sHR (subset_trans (commSg _ (sHL sHs)) sRK). Qed.
Lemma
centrals_nil
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "apply", "commSg", "defG", "forall_inP", "gT", "group", "last", "nilpotent", "sHG", "seq", "subG1", "subsetIP", "subset_trans" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcn0 A : 'L_0(A) = A.
Proof. by []. Qed.
Lemma
lcn0
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcn1 A : 'L_1(A) = A.
Proof. by []. Qed.
Lemma
lcn1
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcnSn n A : 'L_n.+2(A) = [~: 'L_n.+1(A), A].
Proof. by []. Qed.
Lemma
lcnSn
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcnSnS n G : [~: 'L_n(G), G] \subset 'L_n.+1(G).
Proof. by case: n => //; apply: der1_subG. Qed.
Lemma
lcnSnS
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "apply", "der1_subG" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcnE n A : 'L_n.+1(A) = iter n (fun B => [~: B, A]) A.
Proof. by []. Qed.
Lemma
lcnE
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "iter" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcn2 A : 'L_2(A) = A^`(1).
Proof. by []. Qed.
Lemma
lcn2
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcn_group_set n G : group_set 'L_n(G).
Proof. by case: n => [|[|n]]; apply: groupP. Qed.
Lemma
lcn_group_set
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "apply", "groupP", "group_set" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lower_central_at_group n G
:= Group (lcn_group_set n G).
Canonical
lower_central_at_group
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "lcn_group_set" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcn_char n G : 'L_n(G) \char G.
Proof. by case: n; last elim=> [|n IHn]; rewrite ?char_refl ?lcnSn ?charR. Qed.
Lemma
lcn_char
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "char", "charR", "char_refl", "last", "lcnSn" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcn_normal n G : 'L_n(G) <| G.
Proof. exact/char_normal/lcn_char. Qed.
Lemma
lcn_normal
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "char_normal", "lcn_char" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcn_sub n G : 'L_n(G) \subset G.
Proof. exact/char_sub/lcn_char. Qed.
Lemma
lcn_sub
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "char_sub", "lcn_char" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcn_norm n G : G \subset 'N('L_n(G)).
Proof. exact/char_norm/lcn_char. Qed.
Lemma
lcn_norm
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "char_norm", "lcn_char" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcn_subS n G : 'L_n.+1(G) \subset 'L_n(G).
Proof. case: n => // n; rewrite lcnSn commGC commg_subr. by case/andP: (lcn_normal n.+1 G). Qed.
Lemma
lcn_subS
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "commGC", "commg_subr", "lcnSn", "lcn_normal" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcn_normalS n G : 'L_n.+1(G) <| 'L_n(G).
Proof. by apply: normalS (lcn_normal _ _); rewrite (lcn_subS, lcn_sub). Qed.
Lemma
lcn_normalS
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "apply", "lcn_normal", "lcn_sub", "lcn_subS", "normalS" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcn_central n G : 'L_n(G) / 'L_n.+1(G) \subset 'Z(G / 'L_n.+1(G)).
Proof. case: n => [|n]; first by rewrite trivg_quotient sub1G. by rewrite subsetI quotientS ?lcn_sub ?quotient_cents2r. Qed.
Lemma
lcn_central
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "lcn_sub", "quotientS", "quotient_cents2r", "sub1G", "subsetI", "trivg_quotient" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcn_sub_leq m n G : n <= m -> 'L_m(G) \subset 'L_n(G).
Proof. by move/subnK <-; elim: {m}(m - n) => // m; apply: subset_trans (lcn_subS _ _). Qed.
Lemma
lcn_sub_leq
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "apply", "lcn_subS", "subnK", "subset_trans" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcnS n A B : A \subset B -> 'L_n(A) \subset 'L_n(B).
Proof. by case: n => // n sAB; elim: n => // n IHn; rewrite !lcnSn genS ?imset2S. Qed.
Lemma
lcnS
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "genS", "imset2S", "lcnSn" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcn_cprod n A B G : A \* B = G -> 'L_n(A) \* 'L_n(B) = 'L_n(G).
Proof. case: n => // n /cprodP[[H K -> ->{A B}] defG cHK]. have sL := subset_trans (lcn_sub _ _); rewrite cprodE ?(centSS _ _ cHK) ?sL //. symmetry; elim: n => // n; rewrite lcnSn => ->; rewrite commMG /=. by apply: subset_trans (commg_normr _ _); rewrite sL // -defG mulG_subr. rewrite -!(commGC G) -defG -{1}(centC c...
Lemma
lcn_cprod
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "apply", "centC", "centSS", "centsC", "cents_norm", "commG1P", "commGC", "commMG", "commg_normr", "cprodE", "cprodP", "defG", "lcnSn", "lcn_norm", "lcn_sub", "mul1g", "mulG_subr", "normsR", "subset_trans" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcn_dprod n A B G : A \x B = G -> 'L_n(A) \x 'L_n(B) = 'L_n(G).
Proof. move=> defG; have [[K H defA defB] _ _ tiAB] := dprodP defG. rewrite !dprodEcp // in defG *; last exact: lcn_cprod. by rewrite defA defB; apply/trivgP; rewrite -tiAB defA defB setISS ?lcn_sub. Qed.
Lemma
lcn_dprod
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "apply", "defG", "dprodEcp", "dprodP", "last", "lcn_cprod", "lcn_sub", "setISS", "trivgP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
der_cprod n A B G : A \* B = G -> A^`(n) \* B^`(n) = G^`(n).
Proof. by move=> defG; elim: n => {defG}// n; apply: (lcn_cprod 2). Qed.
Lemma
der_cprod
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "apply", "defG", "lcn_cprod" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
der_dprod n A B G : A \x B = G -> A^`(n) \x B^`(n) = G^`(n).
Proof. by move=> defG; elim: n => {defG}// n; apply: (lcn_dprod 2). Qed.
Lemma
der_dprod
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "apply", "defG", "lcn_dprod" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcn_bigcprod n I r P (F : I -> {set gT}) G : \big[cprod/1]_(i <- r | P i) F i = G -> \big[cprod/1]_(i <- r | P i) 'L_n(F i) = 'L_n(G).
Proof. elim/big_rec2: _ G => [_ <- | i A Z _ IH G dG]; first exact/esym/trivgP/lcn_sub. by rewrite -(lcn_cprod n dG); have [[_ H _ dH]] := cprodP dG; rewrite dH (IH H). Qed.
Lemma
lcn_bigcprod
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "big_rec2", "cprod", "cprodP", "gT", "lcn_cprod", "lcn_sub", "trivgP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcn_bigdprod n I r P (F : I -> {set gT}) G : \big[dprod/1]_(i <- r | P i) F i = G -> \big[dprod/1]_(i <- r | P i) 'L_n(F i) = 'L_n(G).
Proof. elim/big_rec2: _ G => [_ <- | i A Z _ IH G dG]; first exact/esym/trivgP/lcn_sub. by rewrite -(lcn_dprod n dG); have [[_ H _ dH]] := dprodP dG; rewrite dH (IH H). Qed.
Lemma
lcn_bigdprod
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "big_rec2", "dprod", "dprodP", "gT", "lcn_dprod", "lcn_sub", "trivgP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
der_bigcprod n I r P (F : I -> {set gT}) G : \big[cprod/1]_(i <- r | P i) F i = G -> \big[cprod/1]_(i <- r | P i) (F i)^`(n) = G^`(n).
Proof. elim/big_rec2: _ G => [_ <- | i A Z _ IH G dG]; first by rewrite gF1. by rewrite -(der_cprod n dG); have [[_ H _ dH]] := cprodP dG; rewrite dH (IH H). Qed.
Lemma
der_bigcprod
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "big_rec2", "cprod", "cprodP", "der_cprod", "gF1", "gT" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
der_bigdprod n I r P (F : I -> {set gT}) G : \big[dprod/1]_(i <- r | P i) F i = G -> \big[dprod/1]_(i <- r | P i) (F i)^`(n) = G^`(n).
Proof. elim/big_rec2: _ G => [_ <- | i A Z _ IH G dG]; first by rewrite gF1. by rewrite -(der_dprod n dG); have [[_ H _ dH]] := dprodP dG; rewrite dH (IH H). Qed.
Lemma
der_bigdprod
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "big_rec2", "der_dprod", "dprod", "dprodP", "gF1", "gT" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
nilpotent_class G : nilpotent G = (nil_class G < #|G|).
Proof. rewrite /nil_class; set s := mkseq _ _. transitivity (1 \in s); last by rewrite -index_mem size_mkseq. apply/idP/mapP=> {s}/= [nilG | [n _ Ln1]]; last first. apply/forall_inP=> H /subsetIP[sHG sHR]. rewrite -subG1 {}Ln1; elim: n => // n IHn. by rewrite (subset_trans sHR) ?commSg. pose m := #|G|.-1; exists ...
Lemma
nilpotent_class
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "apply", "cardG_gt1", "card_le1_trivg", "cards1", "comm1G", "commSg", "eqsVneq", "forall_inP", "index_mem", "last", "lcnSn", "lcn_norm", "lcn_sub", "leqNgt", "leq_trans", "ltnS", "mapP", "mem_iota", "mkseq", "nil_class", "nil_comm_properl", "nilpotent", "prednK", "prope...
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcn_nil_classP n G : nilpotent G -> reflect ('L_n.+1(G) = 1) (nil_class G <= n).
Proof. rewrite nilpotent_class /nil_class; set s := mkseq _ _. set c := index 1 s => lt_c_G; case: leqP => [le_c_n | lt_n_c]. have Lc1: nth 1 s c = 1 by rewrite nth_index // -index_mem size_mkseq. by left; apply/trivgP; rewrite -Lc1 nth_mkseq ?lcn_sub_leq. right; apply/eqP/negPf; rewrite -(before_find 1 lt_n_c) nth...
Lemma
lcn_nil_classP
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "apply", "before_find", "index", "index_mem", "lcn_sub_leq", "leqP", "ltn_trans", "mkseq", "nil_class", "nilpotent", "nilpotent_class", "nth", "nth_index", "nth_mkseq", "size_mkseq", "trivgP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcnP G : reflect (exists n, 'L_n.+1(G) = 1) (nilpotent G).
Proof. apply: (iffP idP) => [nilG | [n Ln1]]. by exists (nil_class G); apply/lcn_nil_classP. apply/forall_inP=> H /subsetIP[sHG sHR]; rewrite -subG1 -{}Ln1. by elim: n => // n IHn; rewrite (subset_trans sHR) ?commSg. Qed.
Lemma
lcnP
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "apply", "commSg", "forall_inP", "lcn_nil_classP", "nil_class", "nilpotent", "sHG", "subG1", "subsetIP", "subset_trans" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelian_nil G : abelian G -> nilpotent G.
Proof. by move=> abG; apply/lcnP; exists 1%N; apply/commG1P. Qed.
Lemma
abelian_nil
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "abelian", "apply", "commG1P", "lcnP", "nilpotent" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
nil_class0 G : (nil_class G == 0) = (G :==: 1).
Proof. apply/idP/eqP=> [nilG | ->]. by apply/(lcn_nil_classP 0); rewrite ?nilpotent_class (eqP nilG) ?cardG_gt0. by rewrite -leqn0; apply/(lcn_nil_classP 0); rewrite ?nilpotent1. Qed.
Lemma
nil_class0
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "apply", "cardG_gt0", "lcn_nil_classP", "leqn0", "nil_class", "nilpotent1", "nilpotent_class" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
nil_class1 G : (nil_class G <= 1) = abelian G.
Proof. have [-> | ntG] := eqsVneq G 1. by rewrite abelian1 leq_eqVlt ltnS leqn0 nil_class0 eqxx orbT. apply/idP/idP=> cGG. apply/commG1P; apply/(lcn_nil_classP 1); rewrite // nilpotent_class. by rewrite (leq_ltn_trans cGG) // cardG_gt1. by apply/(lcn_nil_classP 1); rewrite ?abelian_nil //; apply/commG1P. Qed.
Lemma
nil_class1
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "abelian", "abelian1", "abelian_nil", "apply", "cGG", "cardG_gt1", "commG1P", "eqsVneq", "eqxx", "lcn_nil_classP", "leq_eqVlt", "leq_ltn_trans", "leqn0", "ltnS", "nil_class", "nil_class0", "nilpotent_class" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
cprod_nil A B G : A \* B = G -> nilpotent G = nilpotent A && nilpotent B.
Proof. move=> defG; case/cprodP: defG (defG) => [[H K -> ->{A B}] defG _] defGc. apply/idP/andP=> [nilG | [/lcnP[m LmH1] /lcnP[n LnK1]]]. by rewrite !(nilpotentS _ nilG) // -defG (mulG_subr, mulG_subl). apply/lcnP; exists (m + n.+1); apply/trivgP. case/cprodP: (lcn_cprod (m.+1 + n.+1) defGc) => _ <- _. by rewrite mul...
Lemma
cprod_nil
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "apply", "cprodP", "defG", "lcnP", "lcn_cprod", "lcn_sub_leq", "leq_addl", "leq_addr", "mulG_subG", "mulG_subl", "mulG_subr", "nilpotent", "nilpotentS", "trivgP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulg_nil G H : H \subset 'C(G) -> nilpotent (G * H) = nilpotent G && nilpotent H.
Proof. by move=> cGH; rewrite -(cprod_nil (cprodEY cGH)) /= cent_joinEr. Qed.
Lemma
mulg_nil
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "cent_joinEr", "cprodEY", "cprod_nil", "nilpotent" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
dprod_nil A B G : A \x B = G -> nilpotent G = nilpotent A && nilpotent B.
Proof. by case/dprodP=> [[H K -> ->] <- cHK _]; rewrite mulg_nil. Qed.
Lemma
dprod_nil
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "dprodP", "mulg_nil", "nilpotent" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigdprod_nil I r (P : pred I) (A_ : I -> {set gT}) G : \big[dprod/1]_(i <- r | P i) A_ i = G -> (forall i, P i -> nilpotent (A_ i)) -> nilpotent G.
Proof. move=> defG nilA; elim/big_rec: _ => [|i B Pi nilB] in G defG *. by rewrite -defG nilpotent1. have [[_ H _ defB] _ _ _] := dprodP defG. by rewrite (dprod_nil defG) nilA //= defB nilB. Qed.
Lemma
bigdprod_nil
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "big_rec", "defG", "dprod", "dprodP", "dprod_nil", "gT", "nilpotent", "nilpotent1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''L_' n ( G )"
:= (lower_central_at_group n G) : Group_scope.
Notation
''L_' n ( G )
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "lower_central_at_group" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcn_cont n : GFunctor.continuous (@lower_central_at n).
Proof. case: n => //; elim=> // n IHn g0T h0T H phi. by rewrite !lcnSn morphimR ?lcn_sub // commSg ?IHn. Qed.
Lemma
lcn_cont
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "commSg", "continuous", "lcnSn", "lcn_sub", "lower_central_at", "morphimR" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcn_igFun n
:= [igFun by lcn_sub^~ n & lcn_cont n].
Canonical
lcn_igFun
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "lcn_cont", "lcn_sub" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lcn_gFun n
:= [gFun by lcn_cont n].
Canonical
lcn_gFun
solvable
solvable/nilpotent.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "fintype", "div", "bigop", "prime", "finset", "fingroup", "morphism", "automorphism", "quotient", "commutator", "gproduct", "perm", "gfunctor", "center", "gseries", "cyclic", "finfun" ]
[ "lcn_cont" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d