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Coq-MathComp

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lcn_mgFun n
:= [mgFun by fun _ G H => @lcnS _ n G H].
Canonical
lcn_mgFun
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" ]
[ "lcnS" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ucn_pmap : exists hZ : GFunctor.pmap, @upper_central_at n = hZ.
Proof. elim: n => [|n' [hZ defZ]]; first by exists trivGfun_pgFun. by exists [pgFun of @center %% hZ]; rewrite /= -defZ. Qed.
Lemma
ucn_pmap
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" ]
[ "center", "n'", "pmap", "trivGfun_pgFun", "upper_central_at" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ucn_group_set gT (G : {group gT}) : group_set 'Z_n(G).
Proof. by have [hZ ->] := ucn_pmap; apply: groupP. Qed.
Lemma
ucn_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", "gT", "group", "groupP", "group_set", "ucn_pmap" ]
Now extract all the intermediate facts of the last proof.
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
upper_central_at_group gT G
:= Group (@ucn_group_set gT G).
Canonical
upper_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" ]
[ "gT", "ucn_group_set" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ucn_sub gT (G : {group gT}) : 'Z_n(G) \subset G.
Proof. by have [hZ ->] := ucn_pmap; apply: gFsub. Qed.
Lemma
ucn_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" ]
[ "apply", "gFsub", "gT", "group", "ucn_pmap" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_ucn : GFunctor.pcontinuous (@upper_central_at n).
Proof. by have [hZ ->] := ucn_pmap; apply: pmorphimF. Qed.
Lemma
morphim_ucn
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", "pcontinuous", "pmorphimF", "ucn_pmap", "upper_central_at" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ucn_igFun
:= [igFun by ucn_sub & morphim_ucn].
Canonical
ucn_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" ]
[ "morphim_ucn", "ucn_sub" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ucn_gFun
:= [gFun by morphim_ucn].
Canonical
ucn_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" ]
[ "morphim_ucn" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ucn_pgFun
:= [pgFun by morphim_ucn].
Canonical
ucn_pgFun
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" ]
[ "morphim_ucn" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ucn_char : 'Z_n(G) \char G.
Proof. exact: gFchar. Qed.
Lemma
ucn_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", "gFchar" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ucn_norm : G \subset 'N('Z_n(G)).
Proof. exact: gFnorm. Qed.
Lemma
ucn_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" ]
[ "gFnorm" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ucn_normal : 'Z_n(G) <| G.
Proof. exact: gFnormal. Qed.
Lemma
ucn_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" ]
[ "gFnormal" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''Z_' n ( G )"
:= (upper_central_at_group 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_group" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ucn0 A : 'Z_0(A) = 1.
Proof. by []. Qed.
Lemma
ucn0
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
ucnSn n A : 'Z_n.+1(A) = coset 'Z_n(A) @*^-1 'Z(A / 'Z_n(A)).
Proof. by []. Qed.
Lemma
ucnSn
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" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ucnE n A : 'Z_n(A) = iter n (fun B => coset B @*^-1 'Z(A / B)) 1.
Proof. by []. Qed.
Lemma
ucnE
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
ucn_subS n G : 'Z_n(G) \subset 'Z_n.+1(G).
Proof. by rewrite -{1}['Z_n(G)]ker_coset morphpreS ?sub1G. Qed.
Lemma
ucn_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" ]
[ "ker_coset", "morphpreS", "sub1G" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ucn_sub_geq m n G : n >= m -> 'Z_m(G) \subset 'Z_n(G).
Proof. move/subnK <-; elim: {n}(n - m) => // n IHn. exact: subset_trans (ucn_subS _ _). Qed.
Lemma
ucn_sub_geq
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" ]
[ "subnK", "subset_trans", "ucn_subS" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ucn_central n G : 'Z_n.+1(G) / 'Z_n(G) = 'Z(G / 'Z_n(G)).
Proof. by rewrite ucnSn cosetpreK. Qed.
Lemma
ucn_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" ]
[ "cosetpreK", "ucnSn" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ucn_normalS n G : 'Z_n(G) <| 'Z_n.+1(G).
Proof. by rewrite (normalS _ _ (ucn_normal n G)) ?ucn_subS ?ucn_sub. Qed.
Lemma
ucn_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" ]
[ "normalS", "ucn_normal", "ucn_sub", "ucn_subS" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ucn_comm n G : [~: 'Z_n.+1(G), G] \subset 'Z_n(G).
Proof. rewrite -quotient_cents2 ?normal_norm ?ucn_normal ?ucn_normalS //. by rewrite ucn_central subsetIr. Qed.
Lemma
ucn_comm
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" ]
[ "normal_norm", "quotient_cents2", "subsetIr", "ucn_central", "ucn_normal", "ucn_normalS" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ucn1 G : 'Z_1(G) = 'Z(G).
Proof. apply: (quotient_inj (normal1 _) (normal1 _)). by rewrite /= (ucn_central 0) -injmF ?norms1 ?coset1_injm. Qed.
Lemma
ucn1
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", "coset1_injm", "injmF", "normal1", "norms1", "quotient_inj", "ucn_central" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ucnSnR n G : 'Z_n.+1(G) = [set x in G | [~: [set x], G] \subset 'Z_n(G)].
Proof. (* apply/setP=> x; rewrite inE -(setIidPr (ucn_sub n.+1 G)) inE ucnSn. *) (* FIXME: before, we got a `rewrite inE` right after the apply/setP=> x. * * However, this rewrite unfolds termes to strange internal HB names. * * We fixed the issue by applying the inE more carefully, but the problem * * needs to...
Lemma
ucnSnR
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", "inE", "quotientS", "quotient_cents2", "setIidPr", "setP", "sub1set", "sub_quotient_pre", "subsetI", "subsetP", "ucnSn", "ucn_norm", "ucn_sub" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ucn_cprod n A B G : A \* B = G -> 'Z_n(A) \* 'Z_n(B) = 'Z_n(G).
Proof. case/cprodP=> [[H K -> ->{A B}] mulHK cHK]. elim: n => [|n /cprodP[_ /= defZ cZn]]; first exact: cprod1g. set Z := 'Z_n(G) in defZ cZn; rewrite (ucnSn n G) /= -/Z. have /mulGsubP[nZH nZK]: H * K \subset 'N(Z) by rewrite mulHK gFnorm. have <-: 'Z(H / Z) * 'Z(K / Z) = 'Z(G / Z). by rewrite -mulHK quotientMl // c...
Lemma
ucn_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", "cent_joinEr", "center_prod", "centsC", "cents_norm", "cprod1g", "cprodE", "cprodP", "gFnorm", "gFsub", "gFsub_trans", "gT", "group", "group_modl", "inj_f", "injm_center", "mulGidPl", "mulGsubP", "mulSGid", "mul_subG", "mulgSS", "n_gt0", "n...
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ucn_dprod n A B G : A \x B = G -> 'Z_n(A) \x 'Z_n(B) = 'Z_n(G).
Proof. move=> defG; have [[K H defA defB] _ _ tiAB] := dprodP defG. rewrite !dprodEcp // in defG *; last exact: ucn_cprod. by rewrite defA defB; apply/trivgP; rewrite -tiAB defA defB setISS ?ucn_sub. Qed.
Lemma
ucn_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", "setISS", "trivgP", "ucn_cprod", "ucn_sub" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ucn_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) 'Z_n(F i) = 'Z_n(G).
Proof. elim/big_rec2: _ G => [_ <- | i A Z _ IH G dG]; first by rewrite gF1. by rewrite -(ucn_cprod n dG); have [[_ H _ dH]] := cprodP dG; rewrite dH (IH H). Qed.
Lemma
ucn_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", "gF1", "gT", "ucn_cprod" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ucn_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) 'Z_n(F i) = 'Z_n(G).
Proof. elim/big_rec2: _ G => [_ <- | i A Z _ IH G dG]; first by rewrite gF1. by rewrite -(ucn_dprod n dG); have [[_ H _ dH]] := dprodP dG; rewrite dH (IH H). Qed.
Lemma
ucn_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", "gF1", "gT", "ucn_dprod" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ucn_lcnP n G : ('L_n.+1(G) == 1) = ('Z_n(G) == G).
Proof. rewrite !eqEsubset sub1G ucn_sub /= andbT -(ucn0 G); set i := (n in LHS). have: i + 0 = n by [rewrite addn0]; elim: i 0 => [j <- //|i IHi j]. rewrite addSnnS => /IHi <- {IHi}; rewrite ucnSn lcnSn. rewrite -sub_morphim_pre ?gFsub_trans ?gFnorm_trans // subsetI. by rewrite morphimS ?gFsub // quotient_cents2 ?gFsub...
Lemma
ucn_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" ]
[ "addSnnS", "addn0", "eqEsubset", "gFnorm_trans", "gFsub", "gFsub_trans", "lcnSn", "morphimS", "quotient_cents2", "sub1G", "sub_morphim_pre", "subsetI", "ucn0", "ucnSn", "ucn_sub" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ucnP G : reflect (exists n, 'Z_n(G) = G) (nilpotent G).
Proof. apply: (iffP (lcnP G)) => -[n /eqP-clGn]; by exists n; apply/eqP; rewrite ucn_lcnP in clGn *. Qed.
Lemma
ucnP
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", "lcnP", "nilpotent", "ucn_lcnP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ucn_nil_classP n G : nilpotent G -> reflect ('Z_n(G) = G) (nil_class G <= n).
Proof. move=> nilG; rewrite (sameP (lcn_nil_classP n nilG) eqP) ucn_lcnP; apply: eqP. Qed.
Lemma
ucn_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", "lcn_nil_classP", "nil_class", "nilpotent", "ucn_lcnP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ucn_id n G : 'Z_n('Z_n(G)) = 'Z_n(G).
Proof. exact: gFid. Qed.
Lemma
ucn_id
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" ]
[ "gFid" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ucn_nilpotent n G : nilpotent 'Z_n(G).
Proof. by apply/ucnP; exists n; rewrite ucn_id. Qed.
Lemma
ucn_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" ]
[ "apply", "nilpotent", "ucnP", "ucn_id" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
nil_class_ucn n G : nil_class 'Z_n(G) <= n.
Proof. by apply/ucn_nil_classP; rewrite ?ucn_nilpotent // ucn_id. Qed.
Lemma
nil_class_ucn
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", "nil_class", "ucn_id", "ucn_nil_classP", "ucn_nilpotent" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_lcn n G : G \subset D -> f @* 'L_n(G) = 'L_n(f @* G).
Proof. move=> sHG; case: n => //; elim=> // n IHn. by rewrite !lcnSn -IHn morphimR // (subset_trans _ sHG) // lcn_sub. Qed.
Lemma
morphim_lcn
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" ]
[ "lcnSn", "lcn_sub", "morphimR", "sHG", "subset_trans" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_ucn n G : 'injm f -> G \subset D -> f @* 'Z_n(G) = 'Z_n(f @* G).
Proof. exact: injmF. Qed.
Lemma
injm_ucn
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" ]
[ "injmF" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_nil G : nilpotent G -> nilpotent (f @* G).
Proof. case/ucnP=> n ZnG; apply/ucnP; exists n; apply/eqP. by rewrite eqEsubset ucn_sub /= -{1}ZnG morphim_ucn. Qed.
Lemma
morphim_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", "eqEsubset", "morphim_ucn", "nilpotent", "ucnP", "ucn_sub" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_nil G : 'injm f -> G \subset D -> nilpotent (f @* G) = nilpotent G.
Proof. move=> injf sGD; apply/idP/idP; last exact: morphim_nil. case/ucnP=> n; rewrite -injm_ucn // => /injm_morphim_inj defZ. by apply/ucnP; exists n; rewrite defZ ?gFsub_trans. Qed.
Lemma
injm_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", "gFsub_trans", "injf", "injm_morphim_inj", "injm_ucn", "last", "morphim_nil", "nilpotent", "sGD", "ucnP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
nil_class_morphim G : nilpotent G -> nil_class (f @* G) <= nil_class G.
Proof. move=> nilG; rewrite (sameP (ucn_nil_classP _ (morphim_nil nilG)) eqP) /=. by rewrite eqEsubset ucn_sub -{1}(ucn_nil_classP _ nilG (leqnn _)) morphim_ucn. Qed.
Lemma
nil_class_morphim
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" ]
[ "eqEsubset", "leqnn", "morphim_nil", "morphim_ucn", "nil_class", "nilpotent", "ucn_nil_classP", "ucn_sub" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
nil_class_injm G : 'injm f -> G \subset D -> nil_class (f @* G) = nil_class G.
Proof. move=> injf sGD; case nilG: (nilpotent G). apply/eqP; rewrite eqn_leq nil_class_morphim //. rewrite (sameP (lcn_nil_classP _ nilG) eqP) -subG1. rewrite -(injmSK injf) ?gFsub_trans // morphim1. by rewrite morphim_lcn // (lcn_nil_classP _ _ (leqnn _)) //= injm_nil. transitivity #|G|; apply/eqP; rewrite eqn...
Lemma
nil_class_injm
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", "card_injm", "eqn_leq", "gFsub_trans", "index_size", "injf", "injmSK", "injm_nil", "lcn_nil_classP", "leqNgt", "leq_trans", "leqnn", "morphim1", "morphim_lcn", "nil_class", "nil_class_morphim", "nilpotent", "nilpotent_class", "sGD", "size_mkseq", "subG1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_ucn_add m n G : 'Z_(m + n)(G) / 'Z_n(G) = 'Z_m(G / 'Z_n(G)).
Proof. elim: m => [|m IHm]; first exact: trivg_quotient. apply/setP=> Zx; have [x Nx ->{Zx}] := cosetP Zx. have [sZG nZG] := andP (ucn_normal n G). rewrite (ucnSnR m) inE -!sub1set -morphim_set1 //= -quotientR ?sub1set // -IHm. rewrite !quotientSGK ?(ucn_sub_geq, leq_addl, comm_subG _ nZG, sub1set) //=. by rewrite addS...
Lemma
quotient_ucn_add
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" ]
[ "addSn", "apply", "comm_subG", "cosetP", "inE", "leq_addl", "morphim_set1", "quotientR", "quotientSGK", "setP", "sub1set", "trivg_quotient", "ucnSnR", "ucn_normal", "ucn_sub_geq" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog_nil rT G (L : {group rT}) : G \isog L -> nilpotent G = nilpotent L.
Proof. by case/isogP=> f injf <-; rewrite injm_nil. Qed.
Lemma
isog_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" ]
[ "group", "injf", "injm_nil", "isog", "isogP", "nilpotent" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog_nil_class rT G (L : {group rT}) : G \isog L -> nil_class G = nil_class L.
Proof. by case/isogP=> f injf <-; rewrite nil_class_injm. Qed.
Lemma
isog_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" ]
[ "group", "injf", "isog", "isogP", "nil_class", "nil_class_injm" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_nil G H : nilpotent G -> nilpotent (G / H).
Proof. exact: morphim_nil. Qed.
Lemma
quotient_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" ]
[ "morphim_nil", "nilpotent" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_center_nil G : nilpotent (G / 'Z(G)) = nilpotent G.
Proof. rewrite -ucn1; apply/idP/idP; last exact: quotient_nil. case/ucnP=> c nilGq; apply/ucnP; exists c.+1; have nsZ1G := ucn_normal 1 G. apply: (quotient_inj _ nsZ1G); last by rewrite /= -(addn1 c) quotient_ucn_add. by rewrite (normalS _ _ nsZ1G) ?ucn_sub ?ucn_sub_geq. Qed.
Lemma
quotient_center_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" ]
[ "addn1", "apply", "last", "nilpotent", "normalS", "quotient_inj", "quotient_nil", "quotient_ucn_add", "ucn1", "ucnP", "ucn_normal", "ucn_sub", "ucn_sub_geq" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
nil_class_quotient_center G : nilpotent (G) -> nil_class (G / 'Z(G)) = (nil_class G).-1.
Proof. move=> nilG; have nsZ1G := ucn_normal 1 G. apply/eqP; rewrite -ucn1 eqn_leq; apply/andP; split. apply/ucn_nil_classP; rewrite ?quotient_nil //= -quotient_ucn_add ucn1. by rewrite (ucn_nil_classP _ _ _) ?addn1 ?leqSpred. rewrite -subn1 leq_subLR addnC; apply/ucn_nil_classP => //=. apply: (quotient_inj _ nsZ1G...
Lemma
nil_class_quotient_center
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" ]
[ "addn1", "addnC", "addnS", "apply", "eqn_leq", "leqSpred", "leq_subLR", "nil_class", "nilpotent", "normalS", "quotient_inj", "quotient_nil", "quotient_ucn_add", "split", "subn1", "ucn1", "ucn_nil_classP", "ucn_normal", "ucn_sub", "ucn_sub_geq" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
nilpotent_sub_norm G H : nilpotent G -> H \subset G -> 'N_G(H) \subset H -> G :=: H.
Proof. move=> nilG sHG sNH; apply/eqP; rewrite eqEsubset sHG andbT; apply/negP=> nsGH. have{nsGH} [i sZH []]: exists2 i, 'Z_i(G) \subset H & ~ 'Z_i.+1(G) \subset H. case/ucnP: nilG => n ZnG; rewrite -{}ZnG in nsGH. elim: n => [|i IHi] in nsGH *; first by rewrite sub1G in nsGH. by case sZH: ('Z_i(G) \subset H); [e...
Lemma
nilpotent_sub_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" ]
[ "apply", "commgS", "commg_subr", "eqEsubset", "nilpotent", "sHG", "sZH", "sub1G", "subsetI", "subset_trans", "ucnP", "ucn_comm", "ucn_sub" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
nilpotent_proper_norm G H : nilpotent G -> H \proper G -> H \proper 'N_G(H).
Proof. move=> nilG; rewrite properEneq properE subsetI normG => /andP[neHG sHG]. by rewrite sHG; apply: contra neHG => /(nilpotent_sub_norm nilG)->. Qed.
Lemma
nilpotent_proper_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" ]
[ "apply", "nilpotent", "nilpotent_sub_norm", "normG", "proper", "properE", "properEneq", "sHG", "subsetI" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
nilpotent_subnormal G H : nilpotent G -> H \subset G -> H <|<| G.
Proof. move=> nilG; have [m] := ubnP (#|G| - #|H|). elim: m H => // m IHm H /ltnSE-leGHm sHG. have [->|] := eqVproper sHG; first exact: subnormal_refl. move/(nilpotent_proper_norm nilG); set K := 'N_G(H) => prHK. have snHK: H <|<| K by rewrite normal_subnormal ?normalSG. have sKG: K \subset G by rewrite subsetIl. apply...
Lemma
nilpotent_subnormal
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", "eqVproper", "leq_trans", "ltnSE", "ltn_sub2l", "nilpotent", "nilpotent_proper_norm", "normalSG", "normal_subnormal", "proper_card", "proper_sub_trans", "sHG", "sKG", "subnormal_refl", "subnormal_trans", "subsetIl", "ubnP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
TI_center_nil G H : nilpotent G -> H <| G -> H :&: 'Z(G) = 1 -> H :=: 1.
Proof. move=> nilG /andP[sHG nHG] tiHZ. rewrite -{1}(setIidPl sHG); have{nilG} /ucnP[n <-] := nilG. elim: n => [|n IHn]; apply/trivgP; rewrite ?subsetIr // -tiHZ. rewrite [H :&: 'Z(G)]setIA subsetI setIS ?ucn_sub //= (sameP commG1P trivgP). rewrite -commg_subr commGC in nHG. rewrite -IHn subsetI (subset_trans _ nHG) ?c...
Lemma
TI_center_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", "commG1P", "commGC", "commSg", "commg_subr", "nHG", "nilpotent", "sHG", "setIA", "setIS", "setIidPl", "subsetI", "subsetIl", "subsetIr", "subset_trans", "trivgP", "ucnP", "ucn_comm", "ucn_sub" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
meet_center_nil G H : nilpotent G -> H <| G -> H :!=: 1 -> H :&: 'Z(G) != 1.
Proof. by move=> nilG nsHG; apply: contraNneq => /TI_center_nil->. Qed.
Lemma
meet_center_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" ]
[ "TI_center_nil", "apply", "contraNneq", "nilpotent", "nsHG" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
center_nil_eq1 G : nilpotent G -> ('Z(G) == 1) = (G :==: 1).
Proof. move=> nilG; apply/eqP/eqP=> [Z1 | ->]; last exact: center1. by rewrite (TI_center_nil nilG) // (setIidPr (center_sub G)). Qed.
Lemma
center_nil_eq1
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" ]
[ "TI_center_nil", "apply", "center1", "center_sub", "last", "nilpotent", "setIidPr" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
cyclic_nilpotent_quo_der1_cyclic G : nilpotent G -> cyclic (G / G^`(1)) -> cyclic G.
Proof. move=> nG; rewrite (isog_cyclic (quotient1_isog G)). have [-> // | ntG' cGG'] := (eqVneq G^`(1) 1)%g. suffices: 'L_2(G) \subset G :&: 'L_3(G) by move/(eqfun_inP nG)=> <-. rewrite subsetI lcn_sub /= -quotient_cents2 ?lcn_norm //. apply: cyclic_factor_abelian (lcn_central 2 G) _. by rewrite (isog_cyclic (third_iso...
Lemma
cyclic_nilpotent_quo_der1_cyclic
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", "cyclic", "cyclic_factor_abelian", "eqVneq", "eqfun_inP", "isog_cyclic", "lcn_central", "lcn_norm", "lcn_normal", "lcn_sub", "lcn_subS", "nG", "nilpotent", "quotient1_isog", "quotient_cents2", "subsetI", "third_isog" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
nilpotent_sol G : nilpotent G -> solvable G.
Proof. move=> nilG; apply/forall_inP=> H /subsetIP[sHG sHH']. by rewrite (forall_inP nilG) // subsetI sHG (subset_trans sHH') ?commgS. Qed.
Lemma
nilpotent_sol
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", "forall_inP", "nilpotent", "sHG", "solvable", "subsetI", "subsetIP", "subset_trans" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelian_sol G : abelian G -> solvable G.
Proof. by move/abelian_nil/nilpotent_sol. Qed.
Lemma
abelian_sol
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", "abelian_nil", "nilpotent_sol", "solvable" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
solvable1 : solvable [1 gT].
Proof. exact: abelian_sol (abelian1 gT). Qed.
Lemma
solvable1
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" ]
[ "abelian1", "abelian_sol", "gT", "solvable" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
solvableS G H : H \subset G -> solvable G -> solvable H.
Proof. move=> sHG solG; apply/forall_inP=> K /subsetIP[sKH sKK']. by rewrite (forall_inP solG) // subsetI (subset_trans sKH). Qed.
Lemma
solvableS
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", "forall_inP", "sHG", "solvable", "subsetI", "subsetIP", "subset_trans" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sol_der1_proper G H : solvable G -> H \subset G -> H :!=: 1 -> H^`(1) \proper H.
Proof. move=> solG sHG ntH; rewrite properE comm_subG //; apply: implyP ntH. by have:= forallP solG H; rewrite subsetI sHG implybNN. Qed.
Lemma
sol_der1_proper
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", "comm_subG", "forallP", "proper", "properE", "sHG", "solvable", "subsetI" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
derivedP G : reflect (exists n, G^`(n) = 1) (solvable G).
Proof. apply: (iffP idP) => [solG | [n solGn]]; last first. apply/forall_inP=> H /subsetIP[sHG sHH']. rewrite -subG1 -{}solGn; elim: n => // n IHn. exact: subset_trans sHH' (commgSS _ _). suffices IHn n: #|G^`(n)| <= (#|G|.-1 - n).+1. by exists #|G|.-1; rewrite [G^`(_)]card_le1_trivg ?(leq_trans (IHn _)) ?subnn...
Lemma
derivedP
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", "commG1", "commgSS", "der_sub", "dergSn", "eqVneq", "forall_inP", "last", "leqNgt", "leq_trans", "ltnS", "n'", "prednK", "proper_card", "sHG", "sol_der1_proper", "solvable", "subG1", "subn0", "subnS", "subnn", "subsetI...
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_sol : solvable G -> solvable (f @* G).
Proof. move/(solvableS (subsetIr D G)); case/derivedP=> n Gn1; apply/derivedP. by exists n; rewrite /= -morphimIdom -morphim_der ?subsetIl // Gn1 morphim1. Qed.
Lemma
morphim_sol
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", "derivedP", "morphim1", "morphimIdom", "morphim_der", "solvable", "solvableS", "subsetIl", "subsetIr" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_sol : 'injm f -> G \subset D -> solvable (f @* G) = solvable G.
Proof. move=> injf sGD; apply/idP/idP; last exact: morphim_sol. case/derivedP=> n Gn1; apply/derivedP; exists n; apply/trivgP. by rewrite -(injmSK injf) ?gFsub_trans ?morphim_der // Gn1 morphim1. Qed.
Lemma
injm_sol
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", "derivedP", "gFsub_trans", "injf", "injmSK", "last", "morphim1", "morphim_der", "morphim_sol", "sGD", "solvable", "trivgP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog_sol G (L : {group rT}) : G \isog L -> solvable G = solvable L.
Proof. by case/isogP=> f injf <-; rewrite injm_sol. Qed.
Lemma
isog_sol
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" ]
[ "group", "injf", "injm_sol", "isog", "isogP", "solvable" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_sol G H : solvable G -> solvable (G / H).
Proof. exact: morphim_sol. Qed.
Lemma
quotient_sol
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" ]
[ "morphim_sol", "solvable" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
series_sol G H : H <| G -> solvable G = solvable H && solvable (G / H).
Proof. case/andP=> sHG nHG; apply/idP/andP=> [solG | [solH solGH]]. by rewrite quotient_sol // (solvableS sHG). apply/forall_inP=> K /subsetIP[sKG sK'K]. suffices sKH: K \subset H by rewrite (forall_inP solH) // subsetI sKH. have nHK := subset_trans sKG nHG. rewrite -quotient_sub1 // subG1 (forall_inP solGH) //. by r...
Lemma
series_sol
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", "forall_inP", "morphimR", "morphimS", "nHG", "nHK", "quotient_sol", "quotient_sub1", "sHG", "sKG", "solvable", "solvableS", "subG1", "subsetI", "subsetIP", "subset_trans" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
metacyclic_sol G : metacyclic G -> solvable G.
Proof. case/metacyclicP=> K [cycK nsKG cycGq]. by rewrite (series_sol nsKG) !abelian_sol ?cyclic_abelian. Qed.
Lemma
metacyclic_sol
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_sol", "cyclic_abelian", "metacyclic", "metacyclicP", "nsKG", "series_sol", "solvable" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
setXn_sol n (gT : 'I_n -> finGroupType) (G : forall i, {group gT i}) : (forall i, solvable (G i)) -> solvable (setXn G).
Proof. elim: n => [|n IHn] in gT G * => solG; first by rewrite groupX0 solvable1. pose gT' (i : 'I_n) := gT (lift ord0 i). pose prod_group_gT := [the finGroupType of {dffun forall i, gT i}]. pose prod_group_gT' := [the finGroupType of {dffun forall i, gT' i}]. pose f (x : prod_group_gT) : prod_group_gT' := [ffun i => x...
Lemma
setXn_sol
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" ]
[ "aG", "apply", "dom_ker", "eq_irrelevance", "eq_sym", "ffunE", "ffunP", "first_isog", "forallP", "gT", "group", "groupX0", "i0", "imsetP", "inE", "isog_sol", "ker", "kerP", "ker_normal", "last", "lift", "mker", "morphic", "morphimEdom", "morphim_sol", "morphm", "m...
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
pgroup pi A
:= pi.-nat #|A|.
Definition
pgroup
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "nat", "pi" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
psubgroup pi A B
:= (B \subset A) && pgroup pi B.
Definition
psubgroup
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "pgroup", "pi" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_group A
:= pgroup (pdiv #|A|) A.
Definition
p_group
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "pdiv", "pgroup" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_elt pi x
:= pi.-nat #[x].
Definition
p_elt
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "nat", "pi" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
constt x pi
:= x ^+ (chinese #[x]`_pi #[x]`_pi^' 1 0).
Definition
constt
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "chinese", "pi" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Hall A B
:= (B \subset A) && coprime #|B| #|A : B|.
Definition
Hall
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "coprime" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
pHall pi A B
:= [&& B \subset A, pgroup pi B & pi^'.-nat #|A : B|].
Definition
pHall
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "nat", "pgroup", "pi" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Syl p A
:= [set P : {group gT} | pHall p A P].
Definition
Syl
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "gT", "group", "pHall" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Sylow A B
:= p_group B && Hall A B.
Definition
Sylow
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "Hall", "p_group" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"pi .-group"
:= (pgroup pi) (format "pi .-group") : group_scope.
Notation
pi .-group
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "pgroup", "pi" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"pi .-subgroup ( A )"
:= (psubgroup pi A) (format "pi .-subgroup ( A )") : group_scope.
Notation
pi .-subgroup ( A )
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "pi", "psubgroup" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"pi .-elt"
:= (p_elt pi) (format "pi .-elt") : group_scope.
Notation
pi .-elt
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "p_elt", "pi" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x .`_ pi"
:= (constt x pi) (at level 3, left associativity, format "x .`_ pi") : group_scope.
Notation
x .`_ pi
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "constt", "pi" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"pi .-Hall ( G )"
:= (pHall pi G) (format "pi .-Hall ( G )") : group_scope.
Notation
pi .-Hall ( G )
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "pHall", "pi" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"p .-Sylow ( G )"
:= (nat_pred_of_nat p).-Hall(G) (format "p .-Sylow ( G )") : group_scope.
Notation
p .-Sylow ( G )
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "Hall", "nat_pred_of_nat" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''Syl_' p ( G )"
:= (Syl p G) (p at level 2, format "''Syl_' p ( G )") : group_scope.
Notation
''Syl_' p ( G )
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "Syl" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
trivgVpdiv G : G :=: 1 \/ (exists2 p, prime p & p %| #|G|).
Proof. have [leG1|lt1G] := leqP #|G| 1; first by left; apply: card_le1_trivg. by right; exists (pdiv #|G|); rewrite ?pdiv_dvd ?pdiv_prime. Qed.
Lemma
trivgVpdiv
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "apply", "card_le1_trivg", "leqP", "pdiv", "pdiv_dvd", "pdiv_prime", "prime" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
prime_subgroupVti G H : prime #|G| -> G \subset H \/ H :&: G = 1.
Proof. move=> prG; have [|[p p_pr pG]] := trivgVpdiv (H :&: G); first by right. left; rewrite (sameP setIidPr eqP) eqEcard subsetIr. suffices <-: p = #|G| by rewrite dvdn_leq ?cardG_gt0. by apply/eqP; rewrite -dvdn_prime2 // -(LagrangeI G H) setIC dvdn_mulr. Qed.
Lemma
prime_subgroupVti
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "LagrangeI", "apply", "cardG_gt0", "dvdn_leq", "dvdn_mulr", "dvdn_prime2", "eqEcard", "pG", "p_pr", "prime", "setIC", "setIidPr", "subsetIr", "trivgVpdiv" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
pgroupE pi A : pi.-group A = pi.-nat #|A|.
Proof. by []. Qed.
Lemma
pgroupE
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "group", "nat", "pi" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_pgroup pi rho A : {subset pi <= rho} -> pi.-group A -> rho.-group A.
Proof. by move=> pi_sub_rho; apply: sub_in_pnat (in1W pi_sub_rho). Qed.
Lemma
sub_pgroup
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "apply", "group", "pi", "sub_in_pnat" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_pgroup pi rho A : pi =i rho -> pi.-group A = rho.-group A.
Proof. exact: eq_pnat. Qed.
Lemma
eq_pgroup
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "eq_pnat", "group", "pi" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_p'group pi rho A : pi =i rho -> pi^'.-group A = rho^'.-group A.
Proof. by move/eq_negn; apply: eq_pnat. Qed.
Lemma
eq_p'group
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "apply", "eq_negn", "eq_pnat", "group", "pi" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
pgroupNK pi A : pi^'^'.-group A = pi.-group A.
Proof. exact: pnatNK. Qed.
Lemma
pgroupNK
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "group", "pi", "pnatNK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_pgroup p pi A : p.-group A -> p \in pi -> pi.-group A.
Proof. exact: pi_pnat. Qed.
Lemma
pi_pgroup
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "group", "pi", "pi_pnat" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_p'group p pi A : pi.-group A -> p \in pi^' -> p^'.-group A.
Proof. exact: pi_p'nat. Qed.
Lemma
pi_p'group
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "group", "pi", "pi_p'nat" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi'_p'group p pi A : pi^'.-group A -> p \in pi -> p^'.-group A.
Proof. exact: pi'_p'nat. Qed.
Lemma
pi'_p'group
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "group", "pi", "pi'_p'nat" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
p'groupEpi p G : p^'.-group G = (p \notin \pi(G)).
Proof. exact: p'natEpi (cardG_gt0 G). Qed.
Lemma
p'groupEpi
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "cardG_gt0", "group", "p'natEpi", "pi" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
pgroup_pi G : \pi(G).-group G.
Proof. by rewrite /=; apply: pnat_pi. Qed.
Lemma
pgroup_pi
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "apply", "group", "pi", "pnat_pi" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
partG_eq1 pi G : (#|G|`_pi == 1)%N = pi^'.-group G.
Proof. exact: partn_eq1 (cardG_gt0 G). Qed.
Lemma
partG_eq1
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "cardG_gt0", "group", "partn_eq1", "pi" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
pgroupP pi G : reflect (forall p, prime p -> p %| #|G| -> p \in pi) (pi.-group G).
Proof. exact: pnatP. Qed.
Lemma
pgroupP
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "group", "pi", "pnatP", "prime" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
pgroup1 pi : pi.-group [1 gT].
Proof. by rewrite /pgroup cards1. Qed.
Lemma
pgroup1
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "cards1", "gT", "group", "pgroup", "pi" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
pgroupS pi G H : H \subset G -> pi.-group G -> pi.-group H.
Proof. by move=> sHG; apply: pnat_dvd (cardSg sHG). Qed.
Lemma
pgroupS
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "apply", "cardSg", "group", "pi", "pnat_dvd", "sHG" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
oddSg G H : H \subset G -> odd #|G| -> odd #|H|.
Proof. by rewrite !odd_2'nat; apply: pgroupS. Qed.
Lemma
oddSg
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "apply", "odd", "odd_2'nat", "pgroupS" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
odd_pgroup_odd p G : odd p -> p.-group G -> odd #|G|.
Proof. move=> p_odd pG; rewrite odd_2'nat (pi_pnat pG) // !inE. by case: eqP p_odd => // ->. Qed.
Lemma
odd_pgroup_odd
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "group", "inE", "odd", "odd_2'nat", "pG", "pi_pnat" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_pgroup p G : p.-group G -> #|G| = (p ^ logn p #|G|)%N.
Proof. by move=> pG; rewrite -p_part part_pnat_id. Qed.
Lemma
card_pgroup
solvable
solvable/pgroup.v
[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "prime", "fingroup", "morphism", "gfunctor", "automorphism", "quotient", "action", "gproduct", "cyclic" ]
[ "group", "logn", "pG", "p_part", "part_pnat_id" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d