phanerozoic/Coq-MathComp · Datasets at Hugging Face

statement stringlengths 1 4.33k	proof stringlengths 0 37.9k	type stringclasses 25 values	symbolic_name stringlengths 1 67	library stringclasses 10 values	filename stringclasses 112 values	imports listlengths 2 138	deps listlengths 0 64	docstring stringclasses 798 values	source_url stringclasses 1 value	commit stringclasses 1 value
trX : is_transversal X (HG :* <[g]>) G.		Hypothesis	trX	solvable	solvable/finmodule.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "bigop", "ssralg", "finset", "fingroup", "morphism", "perm", "finalg", "action", "gproduct", "commutator", "cyclic", "GRing.Theory", "Fin...	[ "HG", "is_transversal" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sXG : {subset X <= G}.	Proof. exact/subsetP/(transversal_sub trX). Qed.	Let	sXG	solvable	solvable/finmodule.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "bigop", "ssralg", "finset", "fingroup", "morphism", "perm", "finalg", "action", "gproduct", "commutator", "cyclic", "GRing.Theory", "Fin...	[ "subsetP", "trX", "transversal_sub" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcosets_cycle_transversal : H_g_rcosets @: X = HGg.	Proof. have sHXgHGg x: x \in X -> H_g_rcosets x \in HGg. by move/sXG=> Gx; apply: imset_f; rewrite -rcosetE imset_f. apply/setP=> Hxg; apply/imsetP/idP=> [[x /sHXgHGg HGgHxg -> //] \| HGgHxg]. have [_ /rcosetsP[z Gz ->] ->] := imsetP HGgHxg. pose Hzg := H :* z * <[g]>; pose x := transversal_repr 1 X Hzg. have HGgHzg: ...	Lemma	rcosets_cycle_transversal	solvable	solvable/finmodule.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "bigop", "ssralg", "finset", "fingroup", "morphism", "perm", "finalg", "action", "gproduct", "commutator", "cyclic", "GRing.Theory", "Fin...	[ "HG", "HGg", "H_g_rcosets", "apply", "imsetP", "imset_f", "mem_mulg", "mulsgP", "orbit_act", "rcosetE", "rcosetM", "rcoset_eqP", "rcosetsP", "repr_mem_pblock", "repr_mem_transversal", "sXG", "set11", "setP", "trX", "transversal_repr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
defHgX	:= rcosets_cycle_transversal.	Notation	defHgX	solvable	solvable/finmodule.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "bigop", "ssralg", "finset", "fingroup", "morphism", "perm", "finalg", "action", "gproduct", "commutator", "cyclic", "GRing.Theory", "Fin...	[ "rcosets_cycle_transversal" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injHg: {in X &, injective H_g_rcosets}.	Proof. apply/imset_injP; rewrite defHgX (card_transversal trX) defHGg. by rewrite (card_in_imset injHGg). Qed.	Let	injHg	solvable	solvable/finmodule.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "bigop", "ssralg", "finset", "fingroup", "morphism", "perm", "finalg", "action", "gproduct", "commutator", "cyclic", "GRing.Theory", "Fin...	[ "H_g_rcosets", "apply", "card_in_imset", "card_transversal", "defHGg", "defHgX", "imset_injP", "injHGg", "trX" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sum_index_rcosets_cycle : (\sum_(x in X) n_ x)%N = #\|G : H\|.	Proof. by rewrite [#\|G : H\|](card_partition partHGg) -defHgX big_imset. Qed.	Lemma	sum_index_rcosets_cycle	solvable	solvable/finmodule.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "bigop", "ssralg", "finset", "fingroup", "morphism", "perm", "finalg", "action", "gproduct", "commutator", "cyclic", "GRing.Theory", "Fin...	[ "big_imset", "card_partition", "defHgX", "n_", "partHGg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
transfer_cycle_expansion : transfer g = \sum_(x in X) fmalpha ((g ^+ n_ x) ^ (x^-1)).	Proof. pose Y := \bigcup_(x in X) [set x * g ^+ i \| i : 'I_(n_ x)]. pose rY := transversal_repr 1 Y. pose pcyc x := porbit (actperm 'Rs g) (H :* x). pose traj x := traject (actperm 'Rs g) (H :* x) #\|pcyc x\|. have Hgr_eq x: H_g_rcosets x = pcyc x. by rewrite /H_g_rcosets -orbitRs -porbit_actperm ?inE. have pcyc_eq x: ...	Lemma	transfer_cycle_expansion	solvable	solvable/finmodule.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "bigop", "ssralg", "finset", "fingroup", "morphism", "perm", "finalg", "action", "gproduct", "commutator", "cyclic", "GRing.Theory", "Fin...	[ "Gg", "HG", "H_g_rcosets", "actperm", "actpermE", "add0r", "addSnnS", "addn0", "apply", "big_cons", "big_imset", "big_seq1", "big_uniq", "bigcupP", "bigcupsP", "card_in_imset", "card_transversal", "cards1P", "conjgE", "cover_imset", "defHGg", "defHgX", "def_n", "def_pbl...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
semiregular K H	:= {in H^#, forall x, 'C_K[x] = 1}.	Definition	semiregular	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[]	Corresponds to "H acts on K in a regular manner" in B & G.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
semiprime K H	:= {in H^#, forall x, 'C_K[x] = 'C_K(H)}.	Definition	semiprime	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[]	Corresponds to "H acts on K in a prime manner" in B & G.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normedTI A G L	:= [&& A != set0, trivIset (A :^: G) & 'N_G(A) == L].	Definition	normedTI	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "set0", "trivIset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frobenius_group_with_complement G H	:= (H != G) && normedTI H^# G H.	Definition	Frobenius_group_with_complement	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "normedTI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frobenius_group G	:= [exists H : {group gT}, Frobenius_group_with_complement G H].	Definition	Frobenius_group	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_group_with_complement", "gT", "group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frobenius_group_with_kernel_and_complement G K H	:= (K ><\| H == G) && Frobenius_group_with_complement G H.	Definition	Frobenius_group_with_kernel_and_complement	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_group_with_complement" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frobenius_group_with_kernel G K	:= [exists H : {group gT}, Frobenius_group_with_kernel_and_complement G K H].	Definition	Frobenius_group_with_kernel	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_group_with_kernel_and_complement", "gT", "group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frobenius_action	:= [/\ [faithful G, on S \| to], [transitive G, on S \| to], {in G^#, forall x, #\|'Fix_(S \| to)[x]\| <= 1}, H != 1 & exists2 u, u \in S & H = 'C_G[u \| to]].	Definition	Frobenius_action	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "faithful", "on", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
has_Frobenius_action G H : Prop	:= hasFrobeniusAction sT S to of @Frobenius_action G H sT S to.	Variant	has_Frobenius_action	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_action", "sT", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'Frobenius' G 'with' 'complement' H ]"	:= (Frobenius_group_with_complement G H) (G at level 50, format "[ 'Frobenius' G 'with' 'complement' H ]") : group_scope.	Notation	[ 'Frobenius' G 'with' 'complement' H ]	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_group_with_complement" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'Frobenius' G 'with' 'kernel' K ]"	:= (Frobenius_group_with_kernel G K) (format "[ 'Frobenius' G 'with' 'kernel' K ]") : group_scope.	Notation	[ 'Frobenius' G 'with' 'kernel' K ]	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_group_with_kernel" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'Frobenius' G ]"	:= (Frobenius_group G) (format "[ 'Frobenius' G ]") : group_scope.	Notation	[ 'Frobenius' G ]	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'Frobenius' G = K ><\| H ]"	:= (Frobenius_group_with_kernel_and_complement G K H) (K, H at level 35, format "[ 'Frobenius' G = K ><\| H ]") : group_scope.	Notation	[ 'Frobenius' G = K ><\| H ]	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_group_with_kernel_and_complement" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
semiregular1l H : semiregular 1 H.	Proof. by move=> x _ /=; rewrite setI1g. Qed.	Lemma	semiregular1l	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "semiregular", "setI1g" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
semiregular1r K : semiregular K 1.	Proof. by move=> x; rewrite setDv inE. Qed.	Lemma	semiregular1r	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "inE", "semiregular", "setDv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
semiregular_sym H K : semiregular K H -> semiregular H K.	Proof. move=> regH x /setD1P[ntx Kx]; apply: contraNeq ntx. rewrite -subG1 -setD_eq0 -setIDAC => /set0Pn[y /setIP[Hy cxy]]. by rewrite (sameP eqP set1gP) -(regH y Hy) inE Kx cent1C. Qed.	Lemma	semiregular_sym	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "apply", "cent1C", "contraNeq", "inE", "semiregular", "set0Pn", "set1gP", "setD1P", "setD_eq0", "setIDAC", "setIP", "subG1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
semiregularS K1 K2 A1 A2 : K1 \subset K2 -> A1 \subset A2 -> semiregular K2 A2 -> semiregular K1 A1.	Proof. move=> sK12 sA12 regKA2 x /setD1P[ntx /(subsetP sA12)A2x]. by apply/trivgP; rewrite -(regKA2 x) ?inE ?ntx ?setSI. Qed.	Lemma	semiregularS	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "apply", "inE", "semiregular", "setD1P", "setSI", "subsetP", "trivgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
semiregular_prime H K : semiregular K H -> semiprime K H.	Proof. move=> regH x Hx; apply/eqP; rewrite eqEsubset {1}regH // sub1G. by rewrite -cent_set1 setIS ?centS // sub1set; case/setD1P: Hx. Qed.	Lemma	semiregular_prime	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "apply", "centS", "cent_set1", "eqEsubset", "semiprime", "semiregular", "setD1P", "setIS", "sub1G", "sub1set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
semiprime_regular H K : semiprime K H -> 'C_K(H) = 1 -> semiregular K H.	Proof. by move=> prKH tiKcH x Hx; rewrite prKH. Qed.	Lemma	semiprime_regular	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "semiprime", "semiregular" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
semiprimeS K1 K2 A1 A2 : K1 \subset K2 -> A1 \subset A2 -> semiprime K2 A2 -> semiprime K1 A1.	Proof. move=> sK12 sA12 prKA2 x /setD1P[ntx A1x]. apply/eqP; rewrite eqEsubset andbC -{1}cent_set1 setIS ?centS ?sub1set //=. rewrite -(setIidPl sK12) -!setIA prKA2 ?setIS ?centS //. by rewrite !inE ntx (subsetP sA12). Qed.	Lemma	semiprimeS	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "apply", "centS", "cent_set1", "eqEsubset", "inE", "semiprime", "setD1P", "setIA", "setIS", "setIidPl", "sub1set", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cent_semiprime H K X : semiprime K H -> X \subset H -> X :!=: 1 -> 'C_K(X) = 'C_K(H).	Proof. move=> prKH sXH /trivgPn[x Xx ntx]; apply/eqP. rewrite eqEsubset -{1}(prKH x) ?inE ?(subsetP sXH) ?ntx //=. by rewrite -cent_cycle !setIS ?centS ?cycle_subG. Qed.	Lemma	cent_semiprime	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "apply", "centS", "cent_cycle", "cycle_subG", "eqEsubset", "inE", "semiprime", "setIS", "subsetP", "trivgPn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
stab_semiprime H K X : semiprime K H -> X \subset K -> 'C_H(X) != 1 -> 'C_H(X) = H.	Proof. move=> prKH sXK ntCHX; apply/setIidPl; rewrite centsC -subsetIidl. rewrite -{2}(setIidPl sXK) -setIA -(cent_semiprime prKH _ ntCHX) ?subsetIl //. by rewrite !subsetI subxx sXK centsC subsetIr. Qed.	Lemma	stab_semiprime	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "apply", "cent_semiprime", "centsC", "semiprime", "setIA", "setIidPl", "subsetI", "subsetIidl", "subsetIl", "subsetIr", "subxx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cent_semiregular H K X : semiregular K H -> X \subset H -> X :!=: 1 -> 'C_K(X) = 1.	Proof. move=> regKH sXH /trivgPn[x Xx ntx]; apply/trivgP. rewrite -(regKH x) ?inE ?(subsetP sXH) ?ntx ?setIS //=. by rewrite -cent_cycle centS ?cycle_subG. Qed.	Lemma	cent_semiregular	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "apply", "centS", "cent_cycle", "cycle_subG", "inE", "semiregular", "setIS", "subsetP", "trivgP", "trivgPn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
regular_norm_dvd_pred K H : H \subset 'N(K) -> semiregular K H -> #\|H\| %\| #\|K\|.-1.	Proof. move=> nKH regH; have actsH: [acts H, on K^# \| 'J] by rewrite astabsJ normD1. rewrite (cardsD1 1 K) group1 -(acts_sum_card_orbit actsH) /=. rewrite (eq_bigr (fun _ => #\|H\|)) ?sum_nat_const ?dvdn_mull //. move=> _ /imsetP[x /setIdP[ntx Kx] ->]; rewrite card_orbit astab1J. rewrite ['C_H[x]](trivgP _) ?indexg1 //=....	Lemma	regular_norm_dvd_pred	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "acts_sum_card_orbit", "apply", "astab1J", "astabsJ", "card_orbit", "cardsD1", "cent1C", "dvdn_mull", "eq_bigr", "group1", "imsetP", "inE", "indexg1", "nKH", "normD1", "on", "semiregular", "setIP", "setIdP", "subsetP", "sum_nat_const", "trivgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
regular_norm_coprime K H : H \subset 'N(K) -> semiregular K H -> coprime #\|K\| #\|H\|.	Proof. move=> nKH regH. by rewrite (coprime_dvdr (regular_norm_dvd_pred nKH regH)) ?coprimenP. Qed.	Lemma	regular_norm_coprime	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "coprime", "coprime_dvdr", "coprimenP", "nKH", "regular_norm_dvd_pred", "semiregular" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
semiregularJ K H x : semiregular K H -> semiregular (K :^ x) (H :^ x).	Proof. move=> regH yx; rewrite -conjD1g => /imsetP[y Hy ->]. by rewrite cent1J -conjIg regH ?conjs1g. Qed.	Lemma	semiregularJ	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "cent1J", "conjD1g", "conjIg", "conjs1g", "imsetP", "semiregular" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
semiprimeJ K H x : semiprime K H -> semiprime (K :^ x) (H :^ x).	Proof. move=> prH yx; rewrite -conjD1g => /imsetP[y Hy ->]. by rewrite cent1J centJ -!conjIg prH. Qed.	Lemma	semiprimeJ	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "cent1J", "centJ", "conjD1g", "conjIg", "imsetP", "semiprime" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normedTI_P A G L : reflect [/\ A != set0, L \subset 'N_G(A) & {in G, forall g, ~~ [disjoint A & A :^ g] -> g \in L}] (normedTI A G L).	Proof. apply: (iffP and3P) => [[nzA /trivIsetP tiAG /eqP <-] \| [nzA sLN tiAG]]. split=> // g Gg; rewrite inE Gg (sameP normP eqP) /= eq_sym; apply: contraR. by apply: tiAG; rewrite ?mem_orbit ?orbit_refl. have [/set0Pn[a Aa] /subsetIP[_ nAL]] := (nzA, sLN); split=> //; last first. rewrite eqEsubset sLN andbT; app...	Lemma	normedTI_P	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Gg", "act_inj", "apply", "cardJg", "cards_eq0", "conjIg", "conjsgM", "disjoint", "eqEsubset", "eq_sym", "groupMl", "groupV", "imsetP", "inE", "inj_eq", "last", "mem_orbit", "mulgKV", "nAL", "normP", "normedTI", "orbit_refl", "pred0Pn", "set0", "set0Pn", "setIP", ...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normedTI_memJ_P A G L : reflect [/\ A != set0, L \subset G & {in A & G, forall a g, (a ^ g \in A) = (g \in L)}] (normedTI A G L).	Proof. apply: (iffP normedTI_P) => [[-> /subsetIP[sLG nAL] tiAG] \| [-> sLG tiAG]]. split=> // a g Aa Gg; apply/idP/idP=> [Aag \| Lg]; last first. by rewrite memJ_norm ?(subsetP nAL). by apply/tiAG/pred0Pn=> //; exists (a ^ g)%g; rewrite /= Aag memJ_conjg. split=> // [ \| g Gg /pred0Pn[ag /=]]; last first. by re...	Lemma	normedTI_memJ_P	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Gg", "apply", "imsetP", "inE", "last", "memJ_conjg", "memJ_norm", "nAL", "normedTI", "normedTI_P", "pred0Pn", "set0", "split", "subsetIP", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
partition_class_support A G : A != set0 -> trivIset (A :^: G) -> partition (A :^: G) (class_support A G).	Proof. rewrite /partition cover_imset -class_supportEr eqxx => nzA ->. by apply: contra nzA => /imsetP[x _ /eqP]; rewrite eq_sym -!cards_eq0 cardJg. Qed.	Lemma	partition_class_support	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "apply", "cardJg", "cards_eq0", "class_support", "class_supportEr", "cover_imset", "eq_sym", "eqxx", "imsetP", "partition", "set0", "trivIset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
partition_normedTI A G L : normedTI A G L -> partition (A :^: G) (class_support A G).	Proof. by case/and3P=> ntA tiAG _; apply: partition_class_support. Qed.	Lemma	partition_normedTI	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "apply", "class_support", "normedTI", "partition", "partition_class_support" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_support_normedTI A G L : normedTI A G L -> #\|class_support A G\| = (#\|A\| * #\|G : L\|)%N.	Proof. case/and3P=> ntA tiAG /eqP <-; rewrite -card_conjugates mulnC. apply: card_uniform_partition (partition_class_support ntA tiAG). by move=> _ /imsetP[y _ ->]; rewrite cardJg. Qed.	Lemma	card_support_normedTI	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "apply", "cardJg", "card_conjugates", "card_uniform_partition", "class_support", "imsetP", "mulnC", "normedTI", "partition_class_support" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normedTI_S A B G L : A != set0 -> L \subset 'N(A) -> A \subset B -> normedTI B G L -> normedTI A G L.	Proof. move=> nzA /subsetP nAL /subsetP sAB /normedTI_memJ_P[nzB sLG tiB]. apply/normedTI_memJ_P; split=> // a x Aa Gx. by apply/idP/idP => [Aax \| /nAL/memJ_norm-> //]; rewrite -(tiB a) ?sAB. Qed.	Lemma	normedTI_S	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "apply", "memJ_norm", "nAL", "normedTI", "normedTI_memJ_P", "set0", "split", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cent1_normedTI A G L : normedTI A G L -> {in A, forall x, 'C_G[x] \subset L}.	Proof. case/normedTI_memJ_P=> [_ _ tiAG] x Ax; apply/subsetP=> y /setIP[Gy cxy]. by rewrite -(tiAG x) // /(x ^ y) -(cent1P cxy) mulKg. Qed.	Lemma	cent1_normedTI	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "apply", "cent1P", "mulKg", "normedTI", "normedTI_memJ_P", "setIP", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frobenius_actionP G H : reflect (has_Frobenius_action G H) [Frobenius G with complement H].	Proof. apply: (iffP andP) => [[neqHG] \| [sT S to [ffulG transG regG ntH [u Su defH]]]]. case/normedTI_P=> nzH /subsetIP[sHG _] tiHG. suffices: Frobenius_action G H (rcosets H G) 'Rs by apply: hasFrobeniusAction. pose Hfix x := 'Fix_(rcosets H G \| 'Rs)[x]. have regG: {in G^#, forall x, #\|Hfix x\| <= 1}. move=...	Lemma	Frobenius_actionP	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_action", "actsP", "afix1P", "apply", "astab1P", "astab1Rs", "astab1_act", "astabC", "atransP", "atrans_acts", "cardD1", "card_gt0P", "card_orbit", "cards1", "conjD1g", "conjGid", "conjIg", "conjg_eq1", "conjsgM", "contraNneq", "contraTeq", "defG", "eqEcard", ...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
frobG : [Frobenius G = K ><\| H].		Hypothesis	frobG	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
FrobeniusWker : [Frobenius G with kernel K].	Proof. by apply/existsP; exists H. Qed.	Lemma	FrobeniusWker	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "apply", "existsP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
FrobeniusWcompl : [Frobenius G with complement H].	Proof. by case/andP: frobG. Qed.	Lemma	FrobeniusWcompl	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "frobG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
FrobeniusW : [Frobenius G].	Proof. by apply/existsP; exists H; apply: FrobeniusWcompl. Qed.	Lemma	FrobeniusW	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "FrobeniusWcompl", "apply", "existsP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frobenius_context : [/\ K ><\| H = G, K :!=: 1, H :!=: 1, K \proper G & H \proper G].	Proof. have [/eqP defG neqHG ntH _] := and4P frobG; rewrite setD_eq0 subG1 in ntH. have ntK: K :!=: 1 by apply: contraNneq neqHG => K1; rewrite -defG K1 sdprod1g. rewrite properEcard properEneq neqHG; have /mulG_sub[-> ->] := sdprodW defG. by rewrite -(sdprod_card defG) ltn_Pmulr ?cardG_gt1. Qed.	Lemma	Frobenius_context	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "apply", "cardG_gt1", "contraNneq", "defG", "frobG", "ltn_Pmulr", "mulG_sub", "proper", "properEcard", "properEneq", "sdprod1g", "sdprodW", "sdprod_card", "setD_eq0", "subG1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frobenius_partition : partition (gval K \|: (H^# :^: K)) G.	Proof. have [/eqP defG _ tiHG] := and3P frobG; have [_ tiH1G /eqP defN] := and3P tiHG. have [[_ /mulG_sub[sKG sHG] nKH tiKH] mulHK] := (sdprodP defG, sdprodWC defG). set HG := H^# :^: K; set KHG := _ \|: _. have defHG: HG = H^# :^: G. have: 'C_G[H^# \| 'Js] * K = G by rewrite astab1Js defN mulHK. move/subgroup_transi...	Lemma	Frobenius_partition	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "HG", "Lagrange", "apply", "astab1Js", "atransP", "atrans_orbit", "big_setU1", "cardG_gt0", "card_gt0", "card_support_normedTI", "cardsD1", "class_support_subG", "conj0g", "conjGid", "conjIg", "cover", "defG", "eqEcard", "eq_sym", "eqnP", "frobG", "group1", "imsetP", "i...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frobenius_cent1_ker : {in K^#, forall x, 'C_G[x] \subset K}.	Proof. have [/eqP defG _ /normedTI_memJ_P[_ _ tiHG]] := and3P frobG. move=> x /setD1P[ntx Kx]; have [_ /mulG_sub[sKG _] _ tiKH] := sdprodP defG. have [/eqP <- _ _] := and3P Frobenius_partition; rewrite big_distrl /=. apply/bigcupsP=> _ /setU1P[\|/imsetP[y Ky]] ->; first exact: subsetIl. apply: contraR ntx => /subsetPn[z...	Lemma	Frobenius_cent1_ker	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_partition", "apply", "big_distrl", "bigcupsP", "cent1P", "conjJg", "conjg_eq1", "cxz", "defG", "frobG", "imsetP", "inE", "in_group", "in_set1", "mem_conjg", "mulG_sub", "mulKg", "normedTI_memJ_P", "sKG", "sdprodP", "set1gE", "setD1P", "setU1P", "subsetIl", ...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frobenius_reg_ker : semiregular K H.	Proof. move=> x /setD1P[ntx Hx]. apply/trivgP/subsetP=> y /setIP[Ky cxy]; apply: contraR ntx => nty. have K1y: y \in K^# by rewrite inE nty. have [/eqP/sdprod_context[_ sHG _ _ tiKH] _] := andP frobG. suffices: x \in K :&: H by rewrite tiKH inE. by rewrite inE (subsetP (Frobenius_cent1_ker K1y)) // inE cent1C (subsetP ...	Lemma	Frobenius_reg_ker	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_cent1_ker", "apply", "cent1C", "frobG", "inE", "sHG", "sdprod_context", "semiregular", "setD1P", "setIP", "subsetP", "tiKH", "trivgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frobenius_reg_compl : semiregular H K.	Proof. by apply: semiregular_sym; apply: Frobenius_reg_ker. Qed.	Lemma	Frobenius_reg_compl	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_reg_ker", "apply", "semiregular", "semiregular_sym" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frobenius_dvd_ker1 : #\|H\| %\| #\|K\|.-1.	Proof. apply: regular_norm_dvd_pred Frobenius_reg_ker. by have[/sdprodP[]] := Frobenius_context. Qed.	Lemma	Frobenius_dvd_ker1	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_context", "Frobenius_reg_ker", "apply", "regular_norm_dvd_pred", "sdprodP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltn_odd_Frobenius_ker : odd #\|G\| -> #\|H\|.*2 < #\|K\|.	Proof. move/oddSg=> oddG. have [/sdprodW/mulG_sub[sKG sHG] ntK _ _ _] := Frobenius_context. by rewrite dvdn_double_ltn ?oddG ?cardG_gt1 ?Frobenius_dvd_ker1. Qed.	Lemma	ltn_odd_Frobenius_ker	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_context", "Frobenius_dvd_ker1", "cardG_gt1", "dvdn_double_ltn", "mulG_sub", "odd", "oddSg", "sHG", "sKG", "sdprodW" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frobenius_index_dvd_ker1 : #\|G : K\| %\| #\|K\|.-1.	Proof. have[defG _ _ /andP[sKG _] _] := Frobenius_context. by rewrite -divgS // -(sdprod_card defG) mulKn ?Frobenius_dvd_ker1. Qed.	Lemma	Frobenius_index_dvd_ker1	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_context", "Frobenius_dvd_ker1", "defG", "divgS", "mulKn", "sKG", "sdprod_card" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frobenius_coprime : coprime #\|K\| #\|H\|.	Proof. by rewrite (coprime_dvdr Frobenius_dvd_ker1) ?coprimenP. Qed.	Lemma	Frobenius_coprime	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_dvd_ker1", "coprime", "coprime_dvdr", "coprimenP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frobenius_trivg_cent : 'C_K(H) = 1.	Proof. by apply: (cent_semiregular Frobenius_reg_ker); case: Frobenius_context. Qed.	Lemma	Frobenius_trivg_cent	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_context", "Frobenius_reg_ker", "apply", "cent_semiregular" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frobenius_index_coprime : coprime #\|K\| #\|G : K\|.	Proof. by rewrite (coprime_dvdr Frobenius_index_dvd_ker1) ?coprimenP. Qed.	Lemma	Frobenius_index_coprime	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_index_dvd_ker1", "coprime", "coprime_dvdr", "coprimenP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frobenius_ker_Hall : Hall G K.	Proof. have [_ _ _ /andP[sKG _] _] := Frobenius_context. by rewrite /Hall sKG Frobenius_index_coprime. Qed.	Lemma	Frobenius_ker_Hall	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_context", "Frobenius_index_coprime", "Hall", "sKG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frobenius_compl_Hall : Hall G H.	Proof. have [defG _ _ _ _] := Frobenius_context. by rewrite -(sdprod_Hall defG) Frobenius_ker_Hall. Qed.	Lemma	Frobenius_compl_Hall	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_context", "Frobenius_ker_Hall", "Hall", "defG", "sdprod_Hall" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normedTI_J x A G L : normedTI (A :^ x) (G :^ x) (L :^ x) = normedTI A G L.	Proof. rewrite {1}/normedTI normJ -conjIg -(conj0g x) !(can_eq (conjsgK x)). congr [&& _, _ == _ & _]; rewrite /cover (reindex_inj (@conjsg_inj _ x)). by apply: eq_big => Hy; rewrite ?orbit_conjsg ?cardJg. by rewrite bigcupJ cardJg (eq_bigl _ _ (orbit_conjsg _ _ _ _)). Qed.	Lemma	normedTI_J	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "apply", "bigcupJ", "can_eq", "cardJg", "conj0g", "conjIg", "conjsgK", "conjsg_inj", "cover", "eq_big", "eq_bigl", "normJ", "normedTI", "orbit_conjsg", "reindex_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
FrobeniusJcompl x G H : [Frobenius G :^ x with complement H :^ x] = [Frobenius G with complement H].	Proof. by congr (_ && _); rewrite ?(can_eq (conjsgK x)) // -conjD1g normedTI_J. Qed.	Lemma	FrobeniusJcompl	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "can_eq", "conjD1g", "conjsgK", "normedTI_J" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
FrobeniusJ x G K H : [Frobenius G :^ x = K :^ x ><\| H :^ x] = [Frobenius G = K ><\| H].	Proof. by congr (_ && _); rewrite ?FrobeniusJcompl // -sdprodJ (can_eq (conjsgK x)). Qed.	Lemma	FrobeniusJ	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "FrobeniusJcompl", "can_eq", "conjsgK", "sdprodJ" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
FrobeniusJker x G K : [Frobenius G :^ x with kernel K :^ x] = [Frobenius G with kernel K].	Proof. apply/existsP/existsP=> [] [H]; last by exists (H :^ x)%G; rewrite FrobeniusJ. by rewrite -(conjsgKV x H) FrobeniusJ; exists (H :^ x^-1)%G. Qed.	Lemma	FrobeniusJker	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "FrobeniusJ", "apply", "conjsgKV", "existsP", "last" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
FrobeniusJgroup x G : [Frobenius G :^ x] = [Frobenius G].	Proof. apply/existsP/existsP=> [] [H]. by rewrite -(conjsgKV x H) FrobeniusJcompl; exists (H :^ x^-1)%G. by exists (H :^ x)%G; rewrite FrobeniusJcompl. Qed.	Lemma	FrobeniusJgroup	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "FrobeniusJcompl", "apply", "conjsgKV", "existsP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frobenius_ker_dvd_ker1 G K : [Frobenius G with kernel K] -> #\|G : K\| %\| #\|K\|.-1.	Proof. by case/existsP=> H; apply: Frobenius_index_dvd_ker1. Qed.	Lemma	Frobenius_ker_dvd_ker1	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_index_dvd_ker1", "apply", "existsP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frobenius_ker_coprime G K : [Frobenius G with kernel K] -> coprime #\|K\| #\|G : K\|.	Proof. by case/existsP=> H; apply: Frobenius_index_coprime. Qed.	Lemma	Frobenius_ker_coprime	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_index_coprime", "apply", "coprime", "existsP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frobenius_semiregularP G K H : K ><\| H = G -> K :!=: 1 -> H :!=: 1 -> reflect (semiregular K H) [Frobenius G = K ><\| H].	Proof. move=> defG ntK ntH. apply: (iffP idP) => [\|regG]; first exact: Frobenius_reg_ker. have [nsKG sHG defKH nKH tiKH]:= sdprod_context defG; have [sKG _]:= andP nsKG. apply/and3P; split; first by rewrite defG. by rewrite eqEcard sHG -(sdprod_card defG) -ltnNge ltn_Pmull ?cardG_gt1. apply/normedTI_memJ_P; rewrite s...	Lemma	Frobenius_semiregularP	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_reg_ker", "apply", "cardG_gt1", "cent1C", "cent1P", "commgEr", "commgP", "conjgM", "conjg_eq1", "defG", "eqEcard", "groupJ", "groupM", "groupMl", "groupMr", "groupV", "inE", "in_group", "in_set1", "last", "ltnNge", "ltn_Pmull", "memJ_norm", "mulsgP", "nKH",...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prime_FrobeniusP G K H : K :!=: 1 -> prime #\|H\| -> reflect (K ><\| H = G /\ 'C_K(H) = 1) [Frobenius G = K ><\| H].	Proof. move=> ntK H_pr; have ntH: H :!=: 1 by rewrite -cardG_gt1 prime_gt1. have [defG \| not_sdG] := eqVneq (K ><\| H) G; last first. by apply: (iffP andP) => [] [defG]; rewrite defG ?eqxx in not_sdG. apply: (iffP (Frobenius_semiregularP defG ntK ntH)) => [regH \| [_ regH x]]. split=> //; have [x defH] := cyclicP (pr...	Lemma	prime_FrobeniusP	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_semiregularP", "apply", "cardG_gt1", "centS", "cent_cycle", "cycle_eq1", "cycle_id", "cycle_subG", "cyclicP", "defG", "eqVneq", "eqxx", "inE", "last", "prime", "prime_cyclic", "prime_gt1", "prime_meetG", "setD1P", "setIS", "setIidPr", "split", "trivgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frobenius_subl G K K1 H : K1 :!=: 1 -> K1 \subset K -> H \subset 'N(K1) -> [Frobenius G = K ><\| H] -> [Frobenius K1 <*> H = K1 ><\| H].	Proof. move=> ntK1 sK1K nK1H frobG; have [_ _ ntH _ _] := Frobenius_context frobG. apply/Frobenius_semiregularP=> //. by rewrite sdprodEY ?coprime_TIg ?(coprimeSg sK1K) ?(Frobenius_coprime frobG). by move=> x /(Frobenius_reg_ker frobG) cKx1; apply/trivgP; rewrite -cKx1 setSI. Qed.	Lemma	Frobenius_subl	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_context", "Frobenius_coprime", "Frobenius_reg_ker", "Frobenius_semiregularP", "apply", "coprimeSg", "coprime_TIg", "frobG", "sdprodEY", "setSI", "trivgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frobenius_subr G K H H1 : H1 :!=: 1 -> H1 \subset H -> [Frobenius G = K ><\| H] -> [Frobenius K <*> H1 = K ><\| H1].	Proof. move=> ntH1 sH1H frobG; have [defG ntK _ _ _] := Frobenius_context frobG. apply/Frobenius_semiregularP=> //. have [_ _ /(subset_trans sH1H) nH1K tiHK] := sdprodP defG. by rewrite sdprodEY //; apply/trivgP; rewrite -tiHK setIS. by apply: sub_in1 (Frobenius_reg_ker frobG); apply/subsetP/setSD. Qed.	Lemma	Frobenius_subr	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_context", "Frobenius_reg_ker", "Frobenius_semiregularP", "apply", "defG", "frobG", "sdprodEY", "sdprodP", "setIS", "setSD", "subsetP", "subset_trans", "trivgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frobenius_kerP G K : reflect [/\ K :!=: 1, K \proper G, K <\| G & {in K^#, forall x, 'C_G[x] \subset K}] [Frobenius G with kernel K].	Proof. apply: (iffP existsP) => [[H frobG] \| [ntK ltKG nsKG regK]]. have [/sdprod_context[nsKG _ _ _ _] ntK _ ltKG _] := Frobenius_context frobG. by split=> //; apply: Frobenius_cent1_ker frobG. have /andP[sKG nKG] := nsKG. have hallK: Hall G K. rewrite /Hall sKG //= coprime_sym coprime_pi' //. apply: sub_pgrou...	Lemma	Frobenius_kerP	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_cent1_ker", "Frobenius_context", "Frobenius_semiregularP", "Hall", "Hall_setI_normal", "SchurZassenhaus_split", "Sylow_exists", "apply", "complP", "contraNneq", "coprime_pi'", "coprime_sym", "defG", "existsP", "frobG", "inE", "indexgS", "meet_center_nil", "mulG_sub", ...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
set_Frobenius_compl G K H : K ><\| H = G -> [Frobenius G with kernel K] -> [Frobenius G = K ><\| H].	Proof. move=> defG /Frobenius_kerP[ntK ltKG _ regKG]. apply/Frobenius_semiregularP=> //. by apply: contraTneq ltKG => H_1; rewrite -defG H_1 sdprodg1 properxx. apply: semiregular_sym => y /regKG sCyK. have [_ sHG _ _ tiKH] := sdprod_context defG. by apply/trivgP; rewrite /= -(setIidPr sHG) setIAC -tiKH setSI. Qed.	Lemma	set_Frobenius_compl	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_kerP", "Frobenius_semiregularP", "apply", "contraTneq", "defG", "properxx", "sHG", "sdprod_context", "sdprodg1", "semiregular_sym", "setIAC", "setIidPr", "setSI", "tiKH", "trivgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frobenius_kerS G K G1 : G1 \subset G -> K \proper G1 -> [Frobenius G with kernel K] -> [Frobenius G1 with kernel K].	Proof. move=> sG1G ltKG1 /Frobenius_kerP[ntK _ /andP[_ nKG] regKG]. apply/Frobenius_kerP; rewrite /normal proper_sub // (subset_trans sG1G) //. by split=> // x /regKG; apply: subset_trans; rewrite setSI. Qed.	Lemma	Frobenius_kerS	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_kerP", "G1", "apply", "nKG", "normal", "proper", "proper_sub", "setSI", "split", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frobenius_action_kernel_def G H K sT S to : K ><\| H = G -> @Frobenius_action _ G H sT S to -> K :=: 1 :\|: [set x in G \| 'Fix_(S \| to)[x] == set0].	Proof. move=> defG FrobG. have partG: partition (gval K \|: (H^# :^: K)) G. apply: Frobenius_partition; apply/andP; rewrite defG; split=> //. by apply/Frobenius_actionP; apply: hasFrobeniusAction FrobG. have{FrobG} [ffulG transG regG ntH [u Su defH]]:= FrobG. apply/setP=> x /[!inE]; have [-> \| ntx] := eqVneq; first ...	Lemma	Frobenius_action_kernel_def	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_action", "Frobenius_actionP", "Frobenius_partition", "Sv", "actsP", "apply", "astab1_act", "astabC", "atransP2", "atrans_acts", "big_setU1", "bigcupP", "conjD1g", "conjIg", "conjgKV", "conjs1g", "cover", "cover_partition", "defG", "eqVneq", "group1", "imsetP", ...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frobenius_coprime_quotient (gT : finGroupType) (G K H N : {group gT}) : K ><\| H = G -> N <\| G -> coprime #\|K\| #\|H\| /\ H :!=: 1%g -> N \proper K /\ {in H^#, forall x, 'C_K[x] \subset N} -> [Frobenius G / N = (K / N) ><\| (H / N)]%g.	Proof. move=> defG nsNG [coKH ntH] [ltNK regH]. have [[sNK _] [_ /mulG_sub[sKG sHG] _ _]] := (andP ltNK, sdprodP defG). have [_ nNG] := andP nsNG; have nNH := subset_trans sHG nNG. apply/Frobenius_semiregularP; first exact: quotient_coprime_sdprod. - by rewrite quotient_neq1 ?(normalS _ sKG). - by rewrite -(isog_eq1 (q...	Lemma	Frobenius_coprime_quotient	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Frobenius_semiregularP", "abelian_sol", "apply", "cent_cycle", "coprime", "coprimeSg", "coprime_TIg", "coprimegS", "cycle_abelian", "cycle_subG", "defG", "gT", "group", "isog_eq1", "morphimP", "mulG_sub", "nNG", "nNH", "normalS", "nsNG", "proper", "quotientD1", "quotient...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_Frobenius_compl H sGD injf : [Frobenius G with complement H] -> [Frobenius f @* G with complement f @* H].	Proof. case/andP=> neqGH /normedTI_P[nzH /subsetIP[sHG _] tiHG]. have sHD := subset_trans sHG sGD; have sH1D := subset_trans (subD1set H 1) sHD. apply/andP; rewrite (can_in_eq (injmK injf)) //; split=> //. apply/normedTI_P; rewrite normD1 -injmD1 // -!cards_eq0 card_injm // in nzH *. rewrite subsetI normG morphimS //; ...	Lemma	injm_Frobenius_compl	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "Dx", "apply", "can_in_eq", "card_injm", "cards_eq0", "conj_subG", "injf", "injmD1", "injmI", "injmK", "mem_morphim", "morphim0", "morphimJ", "morphimP", "morphimS", "normD1", "normG", "normedTI_P", "sGD", "sHD", "sHG", "setI_eq0", "split", "subD1set", "subsetI", "s...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_Frobenius H K sGD injf : [Frobenius G = K ><\| H] -> [Frobenius f @* G = f @* K ><\| f @* H].	Proof. case/andP=> /eqP defG frobG. by apply/andP; rewrite (injm_sdprod _ injf defG) // eqxx injm_Frobenius_compl. Qed.	Lemma	injm_Frobenius	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "apply", "defG", "eqxx", "frobG", "injf", "injm_Frobenius_compl", "injm_sdprod", "sGD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_Frobenius_ker K sGD injf : [Frobenius G with kernel K] -> [Frobenius f @* G with kernel f @* K].	Proof. case/existsP=> H frobG; apply/existsP. by exists (f @* H)%G; apply: injm_Frobenius. Qed.	Lemma	injm_Frobenius_ker	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "apply", "existsP", "frobG", "injf", "injm_Frobenius", "sGD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_Frobenius_group sGD injf : [Frobenius G] -> [Frobenius f @* G].	Proof. case/existsP=> H frobG; apply/existsP; exists (f @* H)%G. exact: injm_Frobenius_compl. Qed.	Lemma	injm_Frobenius_group	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "apply", "existsP", "frobG", "injf", "injm_Frobenius_compl", "sGD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Frobenius_Ldiv (gT : finGroupType) (G : {group gT}) n : n %\| #\|G\| -> n %\| #\|'Ldiv_n(G)\|.	Proof. move=> nG; move: {2}_.+1 (ltnSn (#\|G\| %/ n)) => mq. elim: mq => // mq IHm in gT G n nG *; case/dvdnP: nG => q oG. have [q_gt0 n_gt0] : 0 < q /\ 0 < n by apply/andP; rewrite -muln_gt0 -oG. rewrite ltnS oG mulnK // => leqm. have:= q_gt0; rewrite leq_eqVlt => /predU1P[q1 \| lt1q]. rewrite -(mul1n n) q1 -oG (setIid...	Theorem	Frobenius_Ldiv	solvable	solvable/frobenius.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]	[ "G'", "Gauss_dvd", "Gauss_dvdr", "Lagrange", "LagrangeI", "addn1", "apply", "cardG_gt0", "cardSg", "card_quotient", "cardsID", "cent1P", "class_lcoset", "class_refl", "commuteX", "commute_sym", "conjMg", "conjgK", "conjgKV", "conjg_set1", "constt1P", "consttC", "consttJ",...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
object_map	:= forall gT : finGroupType, {set gT} -> {set gT}.	Definition	object_map	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_valued	:= forall gT (G : {group gT}), group_set (F G).	Definition	group_valued	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gT", "group", "group_set" ]	Group closure.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
closed	:= forall gT (G : {group gT}), F G \subset G.	Definition	closed	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gT", "group" ]	Subgroup closure.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
continuous	:= forall gT hT (G : {group gT}) (phi : {morphism G >-> hT}), phi @* F G \subset F (phi @* G).	Definition	continuous	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gT", "group", "morphism" ]	General functoriality, i.e., continuity of the object map	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
iso_continuous	:= forall gT hT (G : {group gT}) (phi : {morphism G >-> hT}), 'injm phi -> phi @* F G \subset F (phi @* G).	Definition	iso_continuous	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gT", "group", "morphism" ]	Functoriality on the Grp groupoid (arrows are restricted to isos).	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
continuous_is_iso_continuous : continuous -> iso_continuous.	Proof. by move=> Fcont gT hT G phi inj_phi; apply: Fcont. Qed.	Lemma	continuous_is_iso_continuous	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "apply", "continuous", "gT", "iso_continuous" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcontinuous	:= forall gT hT (G D : {group gT}) (phi : {morphism D >-> hT}), phi @* F G \subset F (phi @* G).	Definition	pcontinuous	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gT", "group", "morphism" ]	Functoriality on Grp with partial morphisms.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcontinuous_is_continuous : pcontinuous -> continuous.	Proof. by move=> Fcont gT hT G; apply: Fcont. Qed.	Lemma	pcontinuous_is_continuous	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "apply", "continuous", "gT", "pcontinuous" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
hereditary	:= forall gT (H G : {group gT}), H \subset G -> F G :&: H \subset F H.	Definition	hereditary	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gT", "group" ]	Heredity with respect to inclusion	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcontinuous_is_hereditary : pcontinuous -> hereditary.	Proof. move=> Fcont gT H G sHG; rewrite -{2}(setIidPl sHG) setIC. by do 2!rewrite -(morphim_idm (subsetIl H _)) morphimIdom ?Fcont. Qed.	Lemma	pcontinuous_is_hereditary	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gT", "hereditary", "morphimIdom", "morphim_idm", "pcontinuous", "sHG", "setIC", "setIidPl", "subsetIl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
monotonic	:= forall gT (H G : {group gT}), H \subset G -> F H \subset F G.	Definition	monotonic	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "gT", "group" ]	Monotonicity with respect to inclusion	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp : object_map	:= fun gT A => F1 (F2 A).	Definition	comp	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "F1", "F2", "gT", "object_map" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
modulo : object_map	:= fun gT A => coset (F2 A) @*^-1 (F1 (A / (F2 A))).	Definition	modulo	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "F1", "F2", "coset", "gT", "object_map" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
iso_map	:= IsoMap { apply : object_map; _ : group_valued apply; _ : closed apply; _ : iso_continuous apply }.	Structure	iso_map	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "apply", "closed", "group_valued", "iso_continuous", "object_map" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
apply : iso_map >-> object_map.		Coercion	apply	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "iso_map", "object_map" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
map	:= Map { iso_of_map : iso_map; _ : continuous iso_of_map }.	Structure	map	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "continuous", "iso_map", "iso_of_map" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
iso_of_map : map >-> iso_map.		Coercion	iso_of_map	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "iso_map", "map" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pmap	:= Pmap { map_of_pmap : map; _ : hereditary map_of_pmap }.	Structure	pmap	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "hereditary", "map", "map_of_pmap" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_of_pmap : pmap >-> map.		Coercion	map_of_pmap	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "map", "pmap" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mono_map	:= MonoMap { map_of_mono : map; _ : monotonic map_of_mono }.	Structure	mono_map	solvable	solvable/gfunctor.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]	[ "map", "map_of_mono", "monotonic" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

trX : is_transversal X (HG :* <[g]>) G.

Hypothesis

trX

solvable

solvable/finmodule.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "bigop", "ssralg", "finset", "fingroup", "morphism", "perm", "finalg", "action", "gproduct", "commutator", "cyclic", "GRing.Theory", "Fin...

[ "HG", "is_transversal" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sXG : {subset X <= G}.

Proof. exact/subsetP/(transversal_sub trX). Qed.

Let

sXG

solvable

solvable/finmodule.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "bigop", "ssralg", "finset", "fingroup", "morphism", "perm", "finalg", "action", "gproduct", "commutator", "cyclic", "GRing.Theory", "Fin...

[ "subsetP", "trX", "transversal_sub" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcosets_cycle_transversal : H_g_rcosets @: X = HGg.

Proof. have sHXgHGg x: x \in X -> H_g_rcosets x \in HGg. by move/sXG=> Gx; apply: imset_f; rewrite -rcosetE imset_f. apply/setP=> Hxg; apply/imsetP/idP=> [[x /sHXgHGg HGgHxg -> //] | HGgHxg]. have [_ /rcosetsP[z Gz ->] ->] := imsetP HGgHxg. pose Hzg := H :* z * <[g]>; pose x := transversal_repr 1 X Hzg. have HGgHzg: ...

Lemma

rcosets_cycle_transversal

solvable

solvable/finmodule.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "bigop", "ssralg", "finset", "fingroup", "morphism", "perm", "finalg", "action", "gproduct", "commutator", "cyclic", "GRing.Theory", "Fin...

[ "HG", "HGg", "H_g_rcosets", "apply", "imsetP", "imset_f", "mem_mulg", "mulsgP", "orbit_act", "rcosetE", "rcosetM", "rcoset_eqP", "rcosetsP", "repr_mem_pblock", "repr_mem_transversal", "sXG", "set11", "setP", "trX", "transversal_repr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

defHgX

:= rcosets_cycle_transversal.

Notation

defHgX

solvable

solvable/finmodule.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "bigop", "ssralg", "finset", "fingroup", "morphism", "perm", "finalg", "action", "gproduct", "commutator", "cyclic", "GRing.Theory", "Fin...

[ "rcosets_cycle_transversal" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injHg: {in X &, injective H_g_rcosets}.

Proof. apply/imset_injP; rewrite defHgX (card_transversal trX) defHGg. by rewrite (card_in_imset injHGg). Qed.

Let

injHg

solvable

solvable/finmodule.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "bigop", "ssralg", "finset", "fingroup", "morphism", "perm", "finalg", "action", "gproduct", "commutator", "cyclic", "GRing.Theory", "Fin...

[ "H_g_rcosets", "apply", "card_in_imset", "card_transversal", "defHGg", "defHgX", "imset_injP", "injHGg", "trX" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sum_index_rcosets_cycle : (\sum_(x in X) n_ x)%N = #|G : H|.

Proof. by rewrite [#|G : H|](card_partition partHGg) -defHgX big_imset. Qed.

Lemma

sum_index_rcosets_cycle

solvable

solvable/finmodule.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "bigop", "ssralg", "finset", "fingroup", "morphism", "perm", "finalg", "action", "gproduct", "commutator", "cyclic", "GRing.Theory", "Fin...

[ "big_imset", "card_partition", "defHgX", "n_", "partHGg" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

transfer_cycle_expansion : transfer g = \sum_(x in X) fmalpha ((g ^+ n_ x) ^ (x^-1)).

Proof. pose Y := \bigcup_(x in X) [set x * g ^+ i | i : 'I_(n_ x)]. pose rY := transversal_repr 1 Y. pose pcyc x := porbit (actperm 'Rs g) (H :* x). pose traj x := traject (actperm 'Rs g) (H :* x) #|pcyc x|. have Hgr_eq x: H_g_rcosets x = pcyc x. by rewrite /H_g_rcosets -orbitRs -porbit_actperm ?inE. have pcyc_eq x: ...

Lemma

transfer_cycle_expansion

solvable

solvable/finmodule.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "choice", "fintype", "bigop", "ssralg", "finset", "fingroup", "morphism", "perm", "finalg", "action", "gproduct", "commutator", "cyclic", "GRing.Theory", "Fin...

[ "Gg", "HG", "H_g_rcosets", "actperm", "actpermE", "add0r", "addSnnS", "addn0", "apply", "big_cons", "big_imset", "big_seq1", "big_uniq", "bigcupP", "bigcupsP", "card_in_imset", "card_transversal", "cards1P", "conjgE", "cover_imset", "defHGg", "defHgX", "def_n", "def_pbl...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

semiregular K H

:= {in H^#, forall x, 'C_K[x] = 1}.

Definition

semiregular

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[]

Corresponds to "H acts on K in a regular manner" in B & G.

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

semiprime K H

:= {in H^#, forall x, 'C_K[x] = 'C_K(H)}.

Definition

semiprime

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[]

Corresponds to "H acts on K in a prime manner" in B & G.

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normedTI A G L

:= [&& A != set0, trivIset (A :^: G) & 'N_G(A) == L].

Definition

normedTI

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "set0", "trivIset" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frobenius_group_with_complement G H

:= (H != G) && normedTI H^# G H.

Definition

Frobenius_group_with_complement

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "normedTI" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frobenius_group G

:= [exists H : {group gT}, Frobenius_group_with_complement G H].

Definition

Frobenius_group

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_group_with_complement", "gT", "group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frobenius_group_with_kernel_and_complement G K H

:= (K ><| H == G) && Frobenius_group_with_complement G H.

Definition

Frobenius_group_with_kernel_and_complement

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_group_with_complement" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frobenius_group_with_kernel G K

:= [exists H : {group gT}, Frobenius_group_with_kernel_and_complement G K H].

Definition

Frobenius_group_with_kernel

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_group_with_kernel_and_complement", "gT", "group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frobenius_action

:= [/\ [faithful G, on S | to], [transitive G, on S | to], {in G^#, forall x, #|'Fix_(S | to)[x]| <= 1}, H != 1 & exists2 u, u \in S & H = 'C_G[u | to]].

Definition

Frobenius_action

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "faithful", "on", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

has_Frobenius_action G H : Prop

:= hasFrobeniusAction sT S to of @Frobenius_action G H sT S to.

Variant

has_Frobenius_action

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_action", "sT", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'Frobenius' G 'with' 'complement' H ]"

:= (Frobenius_group_with_complement G H) (G at level 50, format "[ 'Frobenius' G 'with' 'complement' H ]") : group_scope.

Notation

[ 'Frobenius' G 'with' 'complement' H ]

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_group_with_complement" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'Frobenius' G 'with' 'kernel' K ]"

:= (Frobenius_group_with_kernel G K) (format "[ 'Frobenius' G 'with' 'kernel' K ]") : group_scope.

Notation

[ 'Frobenius' G 'with' 'kernel' K ]

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_group_with_kernel" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'Frobenius' G ]"

:= (Frobenius_group G) (format "[ 'Frobenius' G ]") : group_scope.

Notation

[ 'Frobenius' G ]

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'Frobenius' G = K ><| H ]"

:= (Frobenius_group_with_kernel_and_complement G K H) (K, H at level 35, format "[ 'Frobenius' G = K ><| H ]") : group_scope.

Notation

[ 'Frobenius' G = K ><| H ]

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_group_with_kernel_and_complement" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

semiregular1l H : semiregular 1 H.

Proof. by move=> x _ /=; rewrite setI1g. Qed.

Lemma

semiregular1l

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "semiregular", "setI1g" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

semiregular1r K : semiregular K 1.

Proof. by move=> x; rewrite setDv inE. Qed.

Lemma

semiregular1r

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "inE", "semiregular", "setDv" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

semiregular_sym H K : semiregular K H -> semiregular H K.

Proof. move=> regH x /setD1P[ntx Kx]; apply: contraNeq ntx. rewrite -subG1 -setD_eq0 -setIDAC => /set0Pn[y /setIP[Hy cxy]]. by rewrite (sameP eqP set1gP) -(regH y Hy) inE Kx cent1C. Qed.

Lemma

semiregular_sym

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "apply", "cent1C", "contraNeq", "inE", "semiregular", "set0Pn", "set1gP", "setD1P", "setD_eq0", "setIDAC", "setIP", "subG1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

semiregularS K1 K2 A1 A2 : K1 \subset K2 -> A1 \subset A2 -> semiregular K2 A2 -> semiregular K1 A1.

Proof. move=> sK12 sA12 regKA2 x /setD1P[ntx /(subsetP sA12)A2x]. by apply/trivgP; rewrite -(regKA2 x) ?inE ?ntx ?setSI. Qed.

Lemma

semiregularS

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "apply", "inE", "semiregular", "setD1P", "setSI", "subsetP", "trivgP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

semiregular_prime H K : semiregular K H -> semiprime K H.

Proof. move=> regH x Hx; apply/eqP; rewrite eqEsubset {1}regH // sub1G. by rewrite -cent_set1 setIS ?centS // sub1set; case/setD1P: Hx. Qed.

Lemma

semiregular_prime

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "apply", "centS", "cent_set1", "eqEsubset", "semiprime", "semiregular", "setD1P", "setIS", "sub1G", "sub1set" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

semiprime_regular H K : semiprime K H -> 'C_K(H) = 1 -> semiregular K H.

Proof. by move=> prKH tiKcH x Hx; rewrite prKH. Qed.

Lemma

semiprime_regular

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "semiprime", "semiregular" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

semiprimeS K1 K2 A1 A2 : K1 \subset K2 -> A1 \subset A2 -> semiprime K2 A2 -> semiprime K1 A1.

Proof. move=> sK12 sA12 prKA2 x /setD1P[ntx A1x]. apply/eqP; rewrite eqEsubset andbC -{1}cent_set1 setIS ?centS ?sub1set //=. rewrite -(setIidPl sK12) -!setIA prKA2 ?setIS ?centS //. by rewrite !inE ntx (subsetP sA12). Qed.

Lemma

semiprimeS

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "apply", "centS", "cent_set1", "eqEsubset", "inE", "semiprime", "setD1P", "setIA", "setIS", "setIidPl", "sub1set", "subsetP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cent_semiprime H K X : semiprime K H -> X \subset H -> X :!=: 1 -> 'C_K(X) = 'C_K(H).

Proof. move=> prKH sXH /trivgPn[x Xx ntx]; apply/eqP. rewrite eqEsubset -{1}(prKH x) ?inE ?(subsetP sXH) ?ntx //=. by rewrite -cent_cycle !setIS ?centS ?cycle_subG. Qed.

Lemma

cent_semiprime

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "apply", "centS", "cent_cycle", "cycle_subG", "eqEsubset", "inE", "semiprime", "setIS", "subsetP", "trivgPn" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

stab_semiprime H K X : semiprime K H -> X \subset K -> 'C_H(X) != 1 -> 'C_H(X) = H.

Proof. move=> prKH sXK ntCHX; apply/setIidPl; rewrite centsC -subsetIidl. rewrite -{2}(setIidPl sXK) -setIA -(cent_semiprime prKH _ ntCHX) ?subsetIl //. by rewrite !subsetI subxx sXK centsC subsetIr. Qed.

Lemma

stab_semiprime

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "apply", "cent_semiprime", "centsC", "semiprime", "setIA", "setIidPl", "subsetI", "subsetIidl", "subsetIl", "subsetIr", "subxx" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cent_semiregular H K X : semiregular K H -> X \subset H -> X :!=: 1 -> 'C_K(X) = 1.

Proof. move=> regKH sXH /trivgPn[x Xx ntx]; apply/trivgP. rewrite -(regKH x) ?inE ?(subsetP sXH) ?ntx ?setIS //=. by rewrite -cent_cycle centS ?cycle_subG. Qed.

Lemma

cent_semiregular

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "apply", "centS", "cent_cycle", "cycle_subG", "inE", "semiregular", "setIS", "subsetP", "trivgP", "trivgPn" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

regular_norm_dvd_pred K H : H \subset 'N(K) -> semiregular K H -> #|H| %| #|K|.-1.

Proof. move=> nKH regH; have actsH: [acts H, on K^# | 'J] by rewrite astabsJ normD1. rewrite (cardsD1 1 K) group1 -(acts_sum_card_orbit actsH) /=. rewrite (eq_bigr (fun _ => #|H|)) ?sum_nat_const ?dvdn_mull //. move=> _ /imsetP[x /setIdP[ntx Kx] ->]; rewrite card_orbit astab1J. rewrite ['C_H[x]](trivgP _) ?indexg1 //=....

Lemma

regular_norm_dvd_pred

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "acts_sum_card_orbit", "apply", "astab1J", "astabsJ", "card_orbit", "cardsD1", "cent1C", "dvdn_mull", "eq_bigr", "group1", "imsetP", "inE", "indexg1", "nKH", "normD1", "on", "semiregular", "setIP", "setIdP", "subsetP", "sum_nat_const", "trivgP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

regular_norm_coprime K H : H \subset 'N(K) -> semiregular K H -> coprime #|K| #|H|.

Proof. move=> nKH regH. by rewrite (coprime_dvdr (regular_norm_dvd_pred nKH regH)) ?coprimenP. Qed.

Lemma

regular_norm_coprime

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "coprime", "coprime_dvdr", "coprimenP", "nKH", "regular_norm_dvd_pred", "semiregular" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

semiregularJ K H x : semiregular K H -> semiregular (K :^ x) (H :^ x).

Proof. move=> regH yx; rewrite -conjD1g => /imsetP[y Hy ->]. by rewrite cent1J -conjIg regH ?conjs1g. Qed.

Lemma

semiregularJ

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "cent1J", "conjD1g", "conjIg", "conjs1g", "imsetP", "semiregular" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

semiprimeJ K H x : semiprime K H -> semiprime (K :^ x) (H :^ x).

Proof. move=> prH yx; rewrite -conjD1g => /imsetP[y Hy ->]. by rewrite cent1J centJ -!conjIg prH. Qed.

Lemma

semiprimeJ

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "cent1J", "centJ", "conjD1g", "conjIg", "imsetP", "semiprime" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normedTI_P A G L : reflect [/\ A != set0, L \subset 'N_G(A) & {in G, forall g, ~~ [disjoint A & A :^ g] -> g \in L}] (normedTI A G L).

Proof. apply: (iffP and3P) => [[nzA /trivIsetP tiAG /eqP <-] | [nzA sLN tiAG]]. split=> // g Gg; rewrite inE Gg (sameP normP eqP) /= eq_sym; apply: contraR. by apply: tiAG; rewrite ?mem_orbit ?orbit_refl. have [/set0Pn[a Aa] /subsetIP[_ nAL]] := (nzA, sLN); split=> //; last first. rewrite eqEsubset sLN andbT; app...

Lemma

normedTI_P

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Gg", "act_inj", "apply", "cardJg", "cards_eq0", "conjIg", "conjsgM", "disjoint", "eqEsubset", "eq_sym", "groupMl", "groupV", "imsetP", "inE", "inj_eq", "last", "mem_orbit", "mulgKV", "nAL", "normP", "normedTI", "orbit_refl", "pred0Pn", "set0", "set0Pn", "setIP", ...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normedTI_memJ_P A G L : reflect [/\ A != set0, L \subset G & {in A & G, forall a g, (a ^ g \in A) = (g \in L)}] (normedTI A G L).

Proof. apply: (iffP normedTI_P) => [[-> /subsetIP[sLG nAL] tiAG] | [-> sLG tiAG]]. split=> // a g Aa Gg; apply/idP/idP=> [Aag | Lg]; last first. by rewrite memJ_norm ?(subsetP nAL). by apply/tiAG/pred0Pn=> //; exists (a ^ g)%g; rewrite /= Aag memJ_conjg. split=> // [ | g Gg /pred0Pn[ag /=]]; last first. by re...

Lemma

normedTI_memJ_P

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Gg", "apply", "imsetP", "inE", "last", "memJ_conjg", "memJ_norm", "nAL", "normedTI", "normedTI_P", "pred0Pn", "set0", "split", "subsetIP", "subsetP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

partition_class_support A G : A != set0 -> trivIset (A :^: G) -> partition (A :^: G) (class_support A G).

Proof. rewrite /partition cover_imset -class_supportEr eqxx => nzA ->. by apply: contra nzA => /imsetP[x _ /eqP]; rewrite eq_sym -!cards_eq0 cardJg. Qed.

Lemma

partition_class_support

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "apply", "cardJg", "cards_eq0", "class_support", "class_supportEr", "cover_imset", "eq_sym", "eqxx", "imsetP", "partition", "set0", "trivIset" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

partition_normedTI A G L : normedTI A G L -> partition (A :^: G) (class_support A G).

Proof. by case/and3P=> ntA tiAG _; apply: partition_class_support. Qed.

Lemma

partition_normedTI

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "apply", "class_support", "normedTI", "partition", "partition_class_support" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_support_normedTI A G L : normedTI A G L -> #|class_support A G| = (#|A| * #|G : L|)%N.

Proof. case/and3P=> ntA tiAG /eqP <-; rewrite -card_conjugates mulnC. apply: card_uniform_partition (partition_class_support ntA tiAG). by move=> _ /imsetP[y _ ->]; rewrite cardJg. Qed.

Lemma

card_support_normedTI

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "apply", "cardJg", "card_conjugates", "card_uniform_partition", "class_support", "imsetP", "mulnC", "normedTI", "partition_class_support" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normedTI_S A B G L : A != set0 -> L \subset 'N(A) -> A \subset B -> normedTI B G L -> normedTI A G L.

Proof. move=> nzA /subsetP nAL /subsetP sAB /normedTI_memJ_P[nzB sLG tiB]. apply/normedTI_memJ_P; split=> // a x Aa Gx. by apply/idP/idP => [Aax | /nAL/memJ_norm-> //]; rewrite -(tiB a) ?sAB. Qed.

Lemma

normedTI_S

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "apply", "memJ_norm", "nAL", "normedTI", "normedTI_memJ_P", "set0", "split", "subsetP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cent1_normedTI A G L : normedTI A G L -> {in A, forall x, 'C_G[x] \subset L}.

Proof. case/normedTI_memJ_P=> [_ _ tiAG] x Ax; apply/subsetP=> y /setIP[Gy cxy]. by rewrite -(tiAG x) // /(x ^ y) -(cent1P cxy) mulKg. Qed.

Lemma

cent1_normedTI

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "apply", "cent1P", "mulKg", "normedTI", "normedTI_memJ_P", "setIP", "subsetP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frobenius_actionP G H : reflect (has_Frobenius_action G H) [Frobenius G with complement H].

Proof. apply: (iffP andP) => [[neqHG] | [sT S to [ffulG transG regG ntH [u Su defH]]]]. case/normedTI_P=> nzH /subsetIP[sHG _] tiHG. suffices: Frobenius_action G H (rcosets H G) 'Rs by apply: hasFrobeniusAction. pose Hfix x := 'Fix_(rcosets H G | 'Rs)[x]. have regG: {in G^#, forall x, #|Hfix x| <= 1}. move=...

Lemma

Frobenius_actionP

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_action", "actsP", "afix1P", "apply", "astab1P", "astab1Rs", "astab1_act", "astabC", "atransP", "atrans_acts", "cardD1", "card_gt0P", "card_orbit", "cards1", "conjD1g", "conjGid", "conjIg", "conjg_eq1", "conjsgM", "contraNneq", "contraTeq", "defG", "eqEcard", ...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

frobG : [Frobenius G = K ><| H].

Hypothesis

frobG

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

FrobeniusWker : [Frobenius G with kernel K].

Proof. by apply/existsP; exists H. Qed.

Lemma

FrobeniusWker

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "apply", "existsP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

FrobeniusWcompl : [Frobenius G with complement H].

Proof. by case/andP: frobG. Qed.

Lemma

FrobeniusWcompl

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "frobG" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

FrobeniusW : [Frobenius G].

Proof. by apply/existsP; exists H; apply: FrobeniusWcompl. Qed.

Lemma

FrobeniusW

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "FrobeniusWcompl", "apply", "existsP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frobenius_context : [/\ K ><| H = G, K :!=: 1, H :!=: 1, K \proper G & H \proper G].

Proof. have [/eqP defG neqHG ntH _] := and4P frobG; rewrite setD_eq0 subG1 in ntH. have ntK: K :!=: 1 by apply: contraNneq neqHG => K1; rewrite -defG K1 sdprod1g. rewrite properEcard properEneq neqHG; have /mulG_sub[-> ->] := sdprodW defG. by rewrite -(sdprod_card defG) ltn_Pmulr ?cardG_gt1. Qed.

Lemma

Frobenius_context

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "apply", "cardG_gt1", "contraNneq", "defG", "frobG", "ltn_Pmulr", "mulG_sub", "proper", "properEcard", "properEneq", "sdprod1g", "sdprodW", "sdprod_card", "setD_eq0", "subG1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frobenius_partition : partition (gval K |: (H^# :^: K)) G.

Proof. have [/eqP defG _ tiHG] := and3P frobG; have [_ tiH1G /eqP defN] := and3P tiHG. have [[_ /mulG_sub[sKG sHG] nKH tiKH] mulHK] := (sdprodP defG, sdprodWC defG). set HG := H^# :^: K; set KHG := _ |: _. have defHG: HG = H^# :^: G. have: 'C_G[H^# | 'Js] * K = G by rewrite astab1Js defN mulHK. move/subgroup_transi...

Lemma

Frobenius_partition

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "HG", "Lagrange", "apply", "astab1Js", "atransP", "atrans_orbit", "big_setU1", "cardG_gt0", "card_gt0", "card_support_normedTI", "cardsD1", "class_support_subG", "conj0g", "conjGid", "conjIg", "cover", "defG", "eqEcard", "eq_sym", "eqnP", "frobG", "group1", "imsetP", "i...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frobenius_cent1_ker : {in K^#, forall x, 'C_G[x] \subset K}.

Proof. have [/eqP defG _ /normedTI_memJ_P[_ _ tiHG]] := and3P frobG. move=> x /setD1P[ntx Kx]; have [_ /mulG_sub[sKG _] _ tiKH] := sdprodP defG. have [/eqP <- _ _] := and3P Frobenius_partition; rewrite big_distrl /=. apply/bigcupsP=> _ /setU1P[|/imsetP[y Ky]] ->; first exact: subsetIl. apply: contraR ntx => /subsetPn[z...

Lemma

Frobenius_cent1_ker

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_partition", "apply", "big_distrl", "bigcupsP", "cent1P", "conjJg", "conjg_eq1", "cxz", "defG", "frobG", "imsetP", "inE", "in_group", "in_set1", "mem_conjg", "mulG_sub", "mulKg", "normedTI_memJ_P", "sKG", "sdprodP", "set1gE", "setD1P", "setU1P", "subsetIl", ...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frobenius_reg_ker : semiregular K H.

Proof. move=> x /setD1P[ntx Hx]. apply/trivgP/subsetP=> y /setIP[Ky cxy]; apply: contraR ntx => nty. have K1y: y \in K^# by rewrite inE nty. have [/eqP/sdprod_context[_ sHG _ _ tiKH] _] := andP frobG. suffices: x \in K :&: H by rewrite tiKH inE. by rewrite inE (subsetP (Frobenius_cent1_ker K1y)) // inE cent1C (subsetP ...

Lemma

Frobenius_reg_ker

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_cent1_ker", "apply", "cent1C", "frobG", "inE", "sHG", "sdprod_context", "semiregular", "setD1P", "setIP", "subsetP", "tiKH", "trivgP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frobenius_reg_compl : semiregular H K.

Proof. by apply: semiregular_sym; apply: Frobenius_reg_ker. Qed.

Lemma

Frobenius_reg_compl

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_reg_ker", "apply", "semiregular", "semiregular_sym" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frobenius_dvd_ker1 : #|H| %| #|K|.-1.

Proof. apply: regular_norm_dvd_pred Frobenius_reg_ker. by have[/sdprodP[]] := Frobenius_context. Qed.

Lemma

Frobenius_dvd_ker1

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_context", "Frobenius_reg_ker", "apply", "regular_norm_dvd_pred", "sdprodP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltn_odd_Frobenius_ker : odd #|G| -> #|H|.*2 < #|K|.

Proof. move/oddSg=> oddG. have [/sdprodW/mulG_sub[sKG sHG] ntK _ _ _] := Frobenius_context. by rewrite dvdn_double_ltn ?oddG ?cardG_gt1 ?Frobenius_dvd_ker1. Qed.

Lemma

ltn_odd_Frobenius_ker

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_context", "Frobenius_dvd_ker1", "cardG_gt1", "dvdn_double_ltn", "mulG_sub", "odd", "oddSg", "sHG", "sKG", "sdprodW" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frobenius_index_dvd_ker1 : #|G : K| %| #|K|.-1.

Proof. have[defG _ _ /andP[sKG _] _] := Frobenius_context. by rewrite -divgS // -(sdprod_card defG) mulKn ?Frobenius_dvd_ker1. Qed.

Lemma

Frobenius_index_dvd_ker1

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_context", "Frobenius_dvd_ker1", "defG", "divgS", "mulKn", "sKG", "sdprod_card" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frobenius_coprime : coprime #|K| #|H|.

Proof. by rewrite (coprime_dvdr Frobenius_dvd_ker1) ?coprimenP. Qed.

Lemma

Frobenius_coprime

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_dvd_ker1", "coprime", "coprime_dvdr", "coprimenP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frobenius_trivg_cent : 'C_K(H) = 1.

Proof. by apply: (cent_semiregular Frobenius_reg_ker); case: Frobenius_context. Qed.

Lemma

Frobenius_trivg_cent

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_context", "Frobenius_reg_ker", "apply", "cent_semiregular" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frobenius_index_coprime : coprime #|K| #|G : K|.

Proof. by rewrite (coprime_dvdr Frobenius_index_dvd_ker1) ?coprimenP. Qed.

Lemma

Frobenius_index_coprime

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_index_dvd_ker1", "coprime", "coprime_dvdr", "coprimenP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frobenius_ker_Hall : Hall G K.

Proof. have [_ _ _ /andP[sKG _] _] := Frobenius_context. by rewrite /Hall sKG Frobenius_index_coprime. Qed.

Lemma

Frobenius_ker_Hall

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_context", "Frobenius_index_coprime", "Hall", "sKG" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frobenius_compl_Hall : Hall G H.

Proof. have [defG _ _ _ _] := Frobenius_context. by rewrite -(sdprod_Hall defG) Frobenius_ker_Hall. Qed.

Lemma

Frobenius_compl_Hall

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_context", "Frobenius_ker_Hall", "Hall", "defG", "sdprod_Hall" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normedTI_J x A G L : normedTI (A :^ x) (G :^ x) (L :^ x) = normedTI A G L.

Proof. rewrite {1}/normedTI normJ -conjIg -(conj0g x) !(can_eq (conjsgK x)). congr [&& _, _ == _ & _]; rewrite /cover (reindex_inj (@conjsg_inj _ x)). by apply: eq_big => Hy; rewrite ?orbit_conjsg ?cardJg. by rewrite bigcupJ cardJg (eq_bigl _ _ (orbit_conjsg _ _ _ _)). Qed.

Lemma

normedTI_J

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "apply", "bigcupJ", "can_eq", "cardJg", "conj0g", "conjIg", "conjsgK", "conjsg_inj", "cover", "eq_big", "eq_bigl", "normJ", "normedTI", "orbit_conjsg", "reindex_inj" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

FrobeniusJcompl x G H : [Frobenius G :^ x with complement H :^ x] = [Frobenius G with complement H].

Proof. by congr (_ && _); rewrite ?(can_eq (conjsgK x)) // -conjD1g normedTI_J. Qed.

Lemma

FrobeniusJcompl

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "can_eq", "conjD1g", "conjsgK", "normedTI_J" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

FrobeniusJ x G K H : [Frobenius G :^ x = K :^ x ><| H :^ x] = [Frobenius G = K ><| H].

Proof. by congr (_ && _); rewrite ?FrobeniusJcompl // -sdprodJ (can_eq (conjsgK x)). Qed.

Lemma

FrobeniusJ

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "FrobeniusJcompl", "can_eq", "conjsgK", "sdprodJ" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

FrobeniusJker x G K : [Frobenius G :^ x with kernel K :^ x] = [Frobenius G with kernel K].

Proof. apply/existsP/existsP=> [] [H]; last by exists (H :^ x)%G; rewrite FrobeniusJ. by rewrite -(conjsgKV x H) FrobeniusJ; exists (H :^ x^-1)%G. Qed.

Lemma

FrobeniusJker

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "FrobeniusJ", "apply", "conjsgKV", "existsP", "last" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

FrobeniusJgroup x G : [Frobenius G :^ x] = [Frobenius G].

Proof. apply/existsP/existsP=> [] [H]. by rewrite -(conjsgKV x H) FrobeniusJcompl; exists (H :^ x^-1)%G. by exists (H :^ x)%G; rewrite FrobeniusJcompl. Qed.

Lemma

FrobeniusJgroup

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "FrobeniusJcompl", "apply", "conjsgKV", "existsP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frobenius_ker_dvd_ker1 G K : [Frobenius G with kernel K] -> #|G : K| %| #|K|.-1.

Proof. by case/existsP=> H; apply: Frobenius_index_dvd_ker1. Qed.

Lemma

Frobenius_ker_dvd_ker1

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_index_dvd_ker1", "apply", "existsP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frobenius_ker_coprime G K : [Frobenius G with kernel K] -> coprime #|K| #|G : K|.

Proof. by case/existsP=> H; apply: Frobenius_index_coprime. Qed.

Lemma

Frobenius_ker_coprime

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_index_coprime", "apply", "coprime", "existsP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frobenius_semiregularP G K H : K ><| H = G -> K :!=: 1 -> H :!=: 1 -> reflect (semiregular K H) [Frobenius G = K ><| H].

Proof. move=> defG ntK ntH. apply: (iffP idP) => [|regG]; first exact: Frobenius_reg_ker. have [nsKG sHG defKH nKH tiKH]:= sdprod_context defG; have [sKG _]:= andP nsKG. apply/and3P; split; first by rewrite defG. by rewrite eqEcard sHG -(sdprod_card defG) -ltnNge ltn_Pmull ?cardG_gt1. apply/normedTI_memJ_P; rewrite s...

Lemma

Frobenius_semiregularP

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_reg_ker", "apply", "cardG_gt1", "cent1C", "cent1P", "commgEr", "commgP", "conjgM", "conjg_eq1", "defG", "eqEcard", "groupJ", "groupM", "groupMl", "groupMr", "groupV", "inE", "in_group", "in_set1", "last", "ltnNge", "ltn_Pmull", "memJ_norm", "mulsgP", "nKH",...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prime_FrobeniusP G K H : K :!=: 1 -> prime #|H| -> reflect (K ><| H = G /\ 'C_K(H) = 1) [Frobenius G = K ><| H].

Proof. move=> ntK H_pr; have ntH: H :!=: 1 by rewrite -cardG_gt1 prime_gt1. have [defG | not_sdG] := eqVneq (K ><| H) G; last first. by apply: (iffP andP) => [] [defG]; rewrite defG ?eqxx in not_sdG. apply: (iffP (Frobenius_semiregularP defG ntK ntH)) => [regH | [_ regH x]]. split=> //; have [x defH] := cyclicP (pr...

Lemma

prime_FrobeniusP

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_semiregularP", "apply", "cardG_gt1", "centS", "cent_cycle", "cycle_eq1", "cycle_id", "cycle_subG", "cyclicP", "defG", "eqVneq", "eqxx", "inE", "last", "prime", "prime_cyclic", "prime_gt1", "prime_meetG", "setD1P", "setIS", "setIidPr", "split", "trivgP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frobenius_subl G K K1 H : K1 :!=: 1 -> K1 \subset K -> H \subset 'N(K1) -> [Frobenius G = K ><| H] -> [Frobenius K1 <*> H = K1 ><| H].

Proof. move=> ntK1 sK1K nK1H frobG; have [_ _ ntH _ _] := Frobenius_context frobG. apply/Frobenius_semiregularP=> //. by rewrite sdprodEY ?coprime_TIg ?(coprimeSg sK1K) ?(Frobenius_coprime frobG). by move=> x /(Frobenius_reg_ker frobG) cKx1; apply/trivgP; rewrite -cKx1 setSI. Qed.

Lemma

Frobenius_subl

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_context", "Frobenius_coprime", "Frobenius_reg_ker", "Frobenius_semiregularP", "apply", "coprimeSg", "coprime_TIg", "frobG", "sdprodEY", "setSI", "trivgP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frobenius_subr G K H H1 : H1 :!=: 1 -> H1 \subset H -> [Frobenius G = K ><| H] -> [Frobenius K <*> H1 = K ><| H1].

Proof. move=> ntH1 sH1H frobG; have [defG ntK _ _ _] := Frobenius_context frobG. apply/Frobenius_semiregularP=> //. have [_ _ /(subset_trans sH1H) nH1K tiHK] := sdprodP defG. by rewrite sdprodEY //; apply/trivgP; rewrite -tiHK setIS. by apply: sub_in1 (Frobenius_reg_ker frobG); apply/subsetP/setSD. Qed.

Lemma

Frobenius_subr

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_context", "Frobenius_reg_ker", "Frobenius_semiregularP", "apply", "defG", "frobG", "sdprodEY", "sdprodP", "setIS", "setSD", "subsetP", "subset_trans", "trivgP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frobenius_kerP G K : reflect [/\ K :!=: 1, K \proper G, K <| G & {in K^#, forall x, 'C_G[x] \subset K}] [Frobenius G with kernel K].

Proof. apply: (iffP existsP) => [[H frobG] | [ntK ltKG nsKG regK]]. have [/sdprod_context[nsKG _ _ _ _] ntK _ ltKG _] := Frobenius_context frobG. by split=> //; apply: Frobenius_cent1_ker frobG. have /andP[sKG nKG] := nsKG. have hallK: Hall G K. rewrite /Hall sKG //= coprime_sym coprime_pi' //. apply: sub_pgrou...

Lemma

Frobenius_kerP

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_cent1_ker", "Frobenius_context", "Frobenius_semiregularP", "Hall", "Hall_setI_normal", "SchurZassenhaus_split", "Sylow_exists", "apply", "complP", "contraNneq", "coprime_pi'", "coprime_sym", "defG", "existsP", "frobG", "inE", "indexgS", "meet_center_nil", "mulG_sub", ...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

set_Frobenius_compl G K H : K ><| H = G -> [Frobenius G with kernel K] -> [Frobenius G = K ><| H].

Proof. move=> defG /Frobenius_kerP[ntK ltKG _ regKG]. apply/Frobenius_semiregularP=> //. by apply: contraTneq ltKG => H_1; rewrite -defG H_1 sdprodg1 properxx. apply: semiregular_sym => y /regKG sCyK. have [_ sHG _ _ tiKH] := sdprod_context defG. by apply/trivgP; rewrite /= -(setIidPr sHG) setIAC -tiKH setSI. Qed.

Lemma

set_Frobenius_compl

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_kerP", "Frobenius_semiregularP", "apply", "contraTneq", "defG", "properxx", "sHG", "sdprod_context", "sdprodg1", "semiregular_sym", "setIAC", "setIidPr", "setSI", "tiKH", "trivgP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frobenius_kerS G K G1 : G1 \subset G -> K \proper G1 -> [Frobenius G with kernel K] -> [Frobenius G1 with kernel K].

Proof. move=> sG1G ltKG1 /Frobenius_kerP[ntK _ /andP[_ nKG] regKG]. apply/Frobenius_kerP; rewrite /normal proper_sub // (subset_trans sG1G) //. by split=> // x /regKG; apply: subset_trans; rewrite setSI. Qed.

Lemma

Frobenius_kerS

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_kerP", "G1", "apply", "nKG", "normal", "proper", "proper_sub", "setSI", "split", "subset_trans" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frobenius_action_kernel_def G H K sT S to : K ><| H = G -> @Frobenius_action _ G H sT S to -> K :=: 1 :|: [set x in G | 'Fix_(S | to)[x] == set0].

Proof. move=> defG FrobG. have partG: partition (gval K |: (H^# :^: K)) G. apply: Frobenius_partition; apply/andP; rewrite defG; split=> //. by apply/Frobenius_actionP; apply: hasFrobeniusAction FrobG. have{FrobG} [ffulG transG regG ntH [u Su defH]]:= FrobG. apply/setP=> x /[!inE]; have [-> | ntx] := eqVneq; first ...

Lemma

Frobenius_action_kernel_def

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_action", "Frobenius_actionP", "Frobenius_partition", "Sv", "actsP", "apply", "astab1_act", "astabC", "atransP2", "atrans_acts", "big_setU1", "bigcupP", "conjD1g", "conjIg", "conjgKV", "conjs1g", "cover", "cover_partition", "defG", "eqVneq", "group1", "imsetP", ...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frobenius_coprime_quotient (gT : finGroupType) (G K H N : {group gT}) : K ><| H = G -> N <| G -> coprime #|K| #|H| /\ H :!=: 1%g -> N \proper K /\ {in H^#, forall x, 'C_K[x] \subset N} -> [Frobenius G / N = (K / N) ><| (H / N)]%g.

Proof. move=> defG nsNG [coKH ntH] [ltNK regH]. have [[sNK _] [_ /mulG_sub[sKG sHG] _ _]] := (andP ltNK, sdprodP defG). have [_ nNG] := andP nsNG; have nNH := subset_trans sHG nNG. apply/Frobenius_semiregularP; first exact: quotient_coprime_sdprod. - by rewrite quotient_neq1 ?(normalS _ sKG). - by rewrite -(isog_eq1 (q...

Lemma

Frobenius_coprime_quotient

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Frobenius_semiregularP", "abelian_sol", "apply", "cent_cycle", "coprime", "coprimeSg", "coprime_TIg", "coprimegS", "cycle_abelian", "cycle_subG", "defG", "gT", "group", "isog_eq1", "morphimP", "mulG_sub", "nNG", "nNH", "normalS", "nsNG", "proper", "quotientD1", "quotient...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_Frobenius_compl H sGD injf : [Frobenius G with complement H] -> [Frobenius f @* G with complement f @* H].

Proof. case/andP=> neqGH /normedTI_P[nzH /subsetIP[sHG _] tiHG]. have sHD := subset_trans sHG sGD; have sH1D := subset_trans (subD1set H 1) sHD. apply/andP; rewrite (can_in_eq (injmK injf)) //; split=> //. apply/normedTI_P; rewrite normD1 -injmD1 // -!cards_eq0 card_injm // in nzH *. rewrite subsetI normG morphimS //; ...

Lemma

injm_Frobenius_compl

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "Dx", "apply", "can_in_eq", "card_injm", "cards_eq0", "conj_subG", "injf", "injmD1", "injmI", "injmK", "mem_morphim", "morphim0", "morphimJ", "morphimP", "morphimS", "normD1", "normG", "normedTI_P", "sGD", "sHD", "sHG", "setI_eq0", "split", "subD1set", "subsetI", "s...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_Frobenius H K sGD injf : [Frobenius G = K ><| H] -> [Frobenius f @* G = f @* K ><| f @* H].

Proof. case/andP=> /eqP defG frobG. by apply/andP; rewrite (injm_sdprod _ injf defG) // eqxx injm_Frobenius_compl. Qed.

Lemma

injm_Frobenius

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "apply", "defG", "eqxx", "frobG", "injf", "injm_Frobenius_compl", "injm_sdprod", "sGD" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_Frobenius_ker K sGD injf : [Frobenius G with kernel K] -> [Frobenius f @* G with kernel f @* K].

Proof. case/existsP=> H frobG; apply/existsP. by exists (f @* H)%G; apply: injm_Frobenius. Qed.

Lemma

injm_Frobenius_ker

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "apply", "existsP", "frobG", "injf", "injm_Frobenius", "sGD" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_Frobenius_group sGD injf : [Frobenius G] -> [Frobenius f @* G].

Proof. case/existsP=> H frobG; apply/existsP; exists (f @* H)%G. exact: injm_Frobenius_compl. Qed.

Lemma

injm_Frobenius_group

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "apply", "existsP", "frobG", "injf", "injm_Frobenius_compl", "sGD" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Frobenius_Ldiv (gT : finGroupType) (G : {group gT}) n : n %| #|G| -> n %| #|'Ldiv_n(G)|.

Proof. move=> nG; move: {2}_.+1 (ltnSn (#|G| %/ n)) => mq. elim: mq => // mq IHm in gT G n nG *; case/dvdnP: nG => q oG. have [q_gt0 n_gt0] : 0 < q /\ 0 < n by apply/andP; rewrite -muln_gt0 -oG. rewrite ltnS oG mulnK // => leqm. have:= q_gt0; rewrite leq_eqVlt => /predU1P[q1 | lt1q]. rewrite -(mul1n n) q1 -oG (setIid...

Theorem

Frobenius_Ldiv

solvable

solvable/frobenius.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "fintype", "bigop", "prime", "finset", "fingroup", "morphism", "perm", "action", "quotient", "gproduct", "cyclic", "center", "pgroup", "nilpotent", "sylow", "hall", "abelian" ]

[ "G'", "Gauss_dvd", "Gauss_dvdr", "Lagrange", "LagrangeI", "addn1", "apply", "cardG_gt0", "cardSg", "card_quotient", "cardsID", "cent1P", "class_lcoset", "class_refl", "commuteX", "commute_sym", "conjMg", "conjgK", "conjgKV", "conjg_set1", "constt1P", "consttC", "consttJ",...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

object_map

:= forall gT : finGroupType, {set gT} -> {set gT}.

Definition

object_map

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group_valued

:= forall gT (G : {group gT}), group_set (F G).

Definition

group_valued

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gT", "group", "group_set" ]

Group closure.

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

closed

:= forall gT (G : {group gT}), F G \subset G.

Definition

closed

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gT", "group" ]

Subgroup closure.

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

continuous

:= forall gT hT (G : {group gT}) (phi : {morphism G >-> hT}), phi @* F G \subset F (phi @* G).

Definition

continuous

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gT", "group", "morphism" ]

General functoriality, i.e., continuity of the object map

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

iso_continuous

:= forall gT hT (G : {group gT}) (phi : {morphism G >-> hT}), 'injm phi -> phi @* F G \subset F (phi @* G).

Definition

iso_continuous

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gT", "group", "morphism" ]

Functoriality on the Grp groupoid (arrows are restricted to isos).

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

continuous_is_iso_continuous : continuous -> iso_continuous.

Proof. by move=> Fcont gT hT G phi inj_phi; apply: Fcont. Qed.

Lemma

continuous_is_iso_continuous

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "apply", "continuous", "gT", "iso_continuous" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcontinuous

:= forall gT hT (G D : {group gT}) (phi : {morphism D >-> hT}), phi @* F G \subset F (phi @* G).

Definition

pcontinuous

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gT", "group", "morphism" ]

Functoriality on Grp with partial morphisms.

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcontinuous_is_continuous : pcontinuous -> continuous.

Proof. by move=> Fcont gT hT G; apply: Fcont. Qed.

Lemma

pcontinuous_is_continuous

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "apply", "continuous", "gT", "pcontinuous" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

hereditary

:= forall gT (H G : {group gT}), H \subset G -> F G :&: H \subset F H.

Definition

hereditary

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gT", "group" ]

Heredity with respect to inclusion

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcontinuous_is_hereditary : pcontinuous -> hereditary.

Proof. move=> Fcont gT H G sHG; rewrite -{2}(setIidPl sHG) setIC. by do 2!rewrite -(morphim_idm (subsetIl H _)) morphimIdom ?Fcont. Qed.

Lemma

pcontinuous_is_hereditary

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gT", "hereditary", "morphimIdom", "morphim_idm", "pcontinuous", "sHG", "setIC", "setIidPl", "subsetIl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

monotonic

:= forall gT (H G : {group gT}), H \subset G -> F H \subset F G.

Definition

monotonic

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "gT", "group" ]

Monotonicity with respect to inclusion

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comp : object_map

:= fun gT A => F1 (F2 A).

Definition

comp

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "F1", "F2", "gT", "object_map" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

modulo : object_map

:= fun gT A => coset (F2 A) @*^-1 (F1 (A / (F2 A))).

Definition

modulo

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "F1", "F2", "coset", "gT", "object_map" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

iso_map

:= IsoMap { apply : object_map; _ : group_valued apply; _ : closed apply; _ : iso_continuous apply }.

Structure

iso_map

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "apply", "closed", "group_valued", "iso_continuous", "object_map" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

apply : iso_map >-> object_map.

Coercion

apply

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "iso_map", "object_map" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

map

:= Map { iso_of_map : iso_map; _ : continuous iso_of_map }.

Structure

map

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "continuous", "iso_map", "iso_of_map" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

iso_of_map : map >-> iso_map.

Coercion

iso_of_map

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "iso_map", "map" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pmap

:= Pmap { map_of_pmap : map; _ : hereditary map_of_pmap }.

Structure

pmap

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "hereditary", "map", "map_of_pmap" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

map_of_pmap : pmap >-> map.

Coercion

map_of_pmap

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "map", "pmap" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mono_map

:= MonoMap { map_of_mono : map; _ : monotonic map_of_mono }.

Structure

mono_map

solvable

solvable/gfunctor.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "fintype", "bigop", "finset", "fingroup", "morphism", "automorphism", "quotient", "gproduct", "GFunctor.Exports" ]

[ "map", "map_of_mono", "monotonic" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d