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
ex_maxgroup : (exists G, gP G) -> {G : {group gT} \| maxgroup G gP}.	Proof. move=> exP; have [A maxA]: {A \| maxgroup A gP}. apply: ex_maxset; case: exP => G gPG. by exists (G : {set gT}); rewrite groupP genGidG. by exists <<A>>%G; rewrite /= gen_set_id; case/andP: (maxsetp maxA). Qed.	Lemma	ex_maxgroup	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "exP", "ex_maxset", "gPG", "gT", "genGidG", "gen_set_id", "group", "groupP", "maxA", "maxgroup", "maxsetp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ex_mingroup : (exists G, gP G) -> {G : {group gT} \| mingroup G gP}.	Proof. move=> exP; have [A minA]: {A \| mingroup A gP}. apply: ex_minset; case: exP => G gPG. by exists (G : {set gT}); rewrite groupP genGidG. by exists <<A>>%G; rewrite /= gen_set_id; case/andP: (minsetp minA). Qed.	Lemma	ex_mingroup	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "exP", "ex_minset", "gPG", "gT", "genGidG", "gen_set_id", "group", "groupP", "minA", "mingroup", "minsetp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mingroupP : reflect (gP G /\ forall H, gP H -> H \subset G -> H :=: G) (mingroup G gP).	Proof. apply: (iffP minsetP); rewrite /= groupP genGidG /= => [] [-> minG]. by split=> // H gPH sGH; apply: minG; rewrite // groupP genGidG. by split=> // A; case/andP=> gA gPA; rewrite -(gen_set_id gA); apply: minG. Qed.	Lemma	mingroupP	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "genGidG", "gen_set_id", "groupP", "mingroup", "minsetP", "sGH", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
maxgroupP : reflect (gP G /\ forall H, gP H -> G \subset H -> H :=: G) (maxgroup G gP).	Proof. apply: (iffP maxsetP); rewrite /= groupP genGidG /= => [] [-> maxG]. by split=> // H gPH sGH; apply: maxG; rewrite // groupP genGidG. by split=> // A; case/andP=> gA gPA; rewrite -(gen_set_id gA); apply: maxG. Qed.	Lemma	maxgroupP	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "genGidG", "gen_set_id", "groupP", "maxgroup", "maxsetP", "sGH", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
maxgroupp : maxgroup G gP -> gP G.	Proof. by case/maxgroupP. Qed.	Lemma	maxgroupp	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "maxgroup", "maxgroupP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mingroupp : mingroup G gP -> gP G.	Proof. by case/mingroupP. Qed.	Lemma	mingroupp	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "mingroup", "mingroupP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gPG : gP G.		Hypothesis	gPG	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
maxgroup_exists : {H : {group gT} \| maxgroup H gP & G \subset H}.	Proof. have [A maxA sGA]: {A \| maxgroup A gP & G \subset A}. by apply: maxset_exists; rewrite groupP genGidG. by exists <<A>>%G; rewrite /= gen_set_id; case/andP: (maxsetp maxA). Qed.	Lemma	maxgroup_exists	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "gT", "genGidG", "gen_set_id", "group", "groupP", "maxA", "maxgroup", "maxset_exists", "maxsetp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mingroup_exists : {H : {group gT} \| mingroup H gP & H \subset G}.	Proof. have [A maxA sGA]: {A \| mingroup A gP & A \subset G}. by apply: minset_exists; rewrite groupP genGidG. by exists <<A>>%G; rewrite /= gen_set_id; case/andP: (minsetp maxA). Qed.	Lemma	mingroup_exists	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "gT", "genGidG", "gen_set_id", "group", "groupP", "maxA", "mingroup", "minset_exists", "minsetp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'max' A 'of' G \| gP ]"	:= (maxgroup A (fun G : {group _} => gP)) : group_scope.	Notation	[ 'max' A 'of' G \| gP ]	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "group", "maxgroup" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'max' G \| gP ]"	:= [max gval G of G \| gP] : group_scope.	Notation	[ 'max' G \| gP ]	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "max" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'max' A 'of' G \| gP & gQ ]"	:= [max A of G \| gP && gQ] : group_scope.	Notation	[ 'max' A 'of' G \| gP & gQ ]	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "max" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'max' G \| gP & gQ ]"	:= [max G \| gP && gQ] : group_scope.	Notation	[ 'max' G \| gP & gQ ]	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "max" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'min' A 'of' G \| gP ]"	:= (mingroup A (fun G : {group _} => gP)) : group_scope.	Notation	[ 'min' A 'of' G \| gP ]	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "group", "mingroup" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'min' G \| gP ]"	:= [min gval G of G \| gP] : group_scope.	Notation	[ 'min' G \| gP ]	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'min' A 'of' G \| gP & gQ ]"	:= [min A of G \| gP && gQ] : group_scope.	Notation	[ 'min' A 'of' G \| gP & gQ ]	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'min' G \| gP & gQ ]"	:= [min G \| gP && gQ] : group_scope.	Notation	[ 'min' G \| gP & gQ ]	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "min" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
partial_product A B	:= if A == 1 then B else if B == 1 then A else if [&& group_set A, group_set B & B \subset 'N(A)] then A * B else set0.	Definition	partial_product	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "group_set", "set0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
semidirect_product A B	:= if A :&: B \subset 1%G then partial_product A B else set0.	Definition	semidirect_product	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "partial_product", "set0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
central_product A B	:= if B \subset 'C(A) then partial_product A B else set0.	Definition	central_product	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "partial_product", "set0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
direct_product A B	:= if A :&: B \subset 1%G then central_product A B else set0.	Definition	direct_product	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "central_product", "set0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
complements_to_in A B	:= [set K : {group gT} \| A :&: K == 1 & A * K == B].	Definition	complements_to_in	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "gT", "group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
splits_over B A	:= complements_to_in A B != set0.	Definition	splits_over	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "complements_to_in", "set0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
remgr A B x	:= repr (A :* x :&: B).	Definition	remgr	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "repr" ]	Product remainder functions -- right variant only.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divgr A B x	:= x * (remgr A B x)^-1.	Definition	divgr	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "remgr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pprod	:= (partial_product _).	Notation	pprod	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "partial_product" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod	:= (semidirect_product _).	Notation	sdprod	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "semidirect_product" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cprod	:= (central_product _).	Notation	cprod	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "central_product" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dprod	:= (direct_product _).	Notation	dprod	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "direct_product" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"G ><\| H"	:= (sdprod G H)%g (at level 40, left associativity) : group_scope.	Notation	G ><\| H	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "sdprod" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"G \* H"	:= (cprod G H)%g (at level 40, left associativity) : group_scope.	Notation	G \* H	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "cprod" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"G \x H"	:= (dprod G H)%g (at level 40, left associativity) : group_scope.	Notation	G \x H	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "dprod" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'complements' 'to' A 'in' B ]"	:= (complements_to_in A B) (format "[ 'complements' 'to' A 'in' B ]") : group_scope.	Notation	[ 'complements' 'to' A 'in' B ]	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "complements_to_in" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'splits' B , 'over' A ]"	:= (splits_over B A) (format "[ 'splits' B , 'over' A ]") : group_scope.	Notation	[ 'splits' B , 'over' A ]	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "splits_over" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pprod	:= (partial_product gT).	Notation	pprod	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "gT", "partial_product" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod	:= (semidirect_product gT) (only parsing).	Notation	sdprod	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "gT", "semidirect_product" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cprod	:= (central_product gT) (only parsing).	Notation	cprod	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "central_product", "gT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dprod	:= (direct_product gT) (only parsing).	Notation	dprod	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "direct_product", "gT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pprod1g : left_id 1 pprod.	Proof. by move=> A; rewrite /pprod eqxx. Qed.	Lemma	pprod1g	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "eqxx", "pprod" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pprodg1 : right_id 1 pprod.	Proof. by move=> A; rewrite /pprod eqxx; case: eqP. Qed.	Lemma	pprodg1	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "eqxx", "pprod" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
are_groups A B : Prop	:= AreGroups K H of A = K & B = H.	Variant	are_groups	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_not0 G : set0 <> G.	Proof. by move/setP/(_ 1); rewrite inE group1. Qed.	Lemma	group_not0	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "group1", "inE", "set0", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulg0 : right_zero (@set0 gT) mul.	Proof. by move=> A; apply/setP=> x; rewrite inE; apply/imset2P=> [[y z]]; rewrite inE. Qed.	Lemma	mulg0	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "gT", "imset2P", "inE", "mul", "set0", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul0g : left_zero (@set0 gT) mul.	Proof. by move=> A; apply/setP=> x; rewrite inE; apply/imset2P=> [[y z]]; rewrite inE. Qed.	Lemma	mul0g	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "gT", "imset2P", "inE", "mul", "set0", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pprodP A B G : pprod A B = G -> [/\ are_groups A B, A * B = G & B \subset 'N(A)].	Proof. have Gnot0 := @group_not0 G; rewrite /pprod; do 2?case: eqP => [-> ->\| _]. - by rewrite mul1g norms1; split; first exists 1%G G. - by rewrite mulg1 sub1G; split; first exists G 1%G. by case: and3P => // [[gA gB ->]]; split; first exists (Group gA) (Group gB). Qed.	Lemma	pprodP	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "are_groups", "group_not0", "mul1g", "mulg1", "norms1", "pprod", "split", "sub1G" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pprodE K H : H \subset 'N(K) -> pprod K H = K * H.	Proof. move=> nKH; rewrite /pprod nKH !groupP /=. by do 2?case: eqP => [-> \| _]; rewrite ?mulg1 ?mul1g. Qed.	Lemma	pprodE	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "groupP", "mul1g", "mulg1", "nKH", "pprod" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pprodEY K H : H \subset 'N(K) -> pprod K H = K <*> H.	Proof. by move=> nKH; rewrite pprodE ?norm_joinEr. Qed.	Lemma	pprodEY	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "nKH", "norm_joinEr", "pprod", "pprodE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pprodW A B G : pprod A B = G -> A * B = G.	Proof. by case/pprodP. Qed.	Lemma	pprodW	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "pprod", "pprodP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pprodWC A B G : pprod A B = G -> B * A = G.	Proof. by case/pprodP=> _ <- /normC. Qed.	Lemma	pprodWC	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "normC", "pprod", "pprodP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pprodWY A B G : pprod A B = G -> A <*> B = G.	Proof. by case/pprodP=> [[K H -> ->] <- /norm_joinEr]. Qed.	Lemma	pprodWY	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "norm_joinEr", "pprod", "pprodP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pprodJ A B x : pprod A B :^ x = pprod (A :^ x) (B :^ x).	Proof. rewrite /pprod !conjsg_eq1 !group_setJ normJ conjSg -conjsMg. by do 3?case: ifP => // _; apply: conj0g. Qed.	Lemma	pprodJ	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "conj0g", "conjSg", "conjsMg", "conjsg_eq1", "group_setJ", "normJ", "pprod" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
remgrMl K B x y : y \in K -> remgr K B (y * x) = remgr K B x.	Proof. by move=> Ky; rewrite {1}/remgr rcosetM rcoset_id. Qed.	Lemma	remgrMl	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "rcosetM", "rcoset_id", "remgr" ]	Properties of the remainders	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
remgrP K B x : (remgr K B x \in K :* x :&: B) = (x \in K * B).	Proof. set y := _ x; apply/idP/mulsgP=> [\|[g b Kg Bb x_gb]]. rewrite inE rcoset_sym mem_rcoset => /andP[Kxy' By]. by exists (x * y^-1) y; rewrite ?mulgKV. by apply: (mem_repr b); rewrite inE rcoset_sym mem_rcoset x_gb mulgK Kg. Qed.	Lemma	remgrP	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "inE", "mem_rcoset", "mem_repr", "mulgK", "mulgKV", "mulsgP", "rcoset_sym", "remgr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
remgr1 K H x : x \in K -> remgr K H x = 1.	Proof. by move=> Kx; rewrite /remgr rcoset_id ?repr_group. Qed.	Lemma	remgr1	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "rcoset_id", "remgr", "repr_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divgr_eq A B x : x = divgr A B x * remgr A B x.	Proof. by rewrite mulgKV. Qed.	Lemma	divgr_eq	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "divgr", "mulgKV", "remgr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divgrMl K B x y : x \in K -> divgr K B (x * y) = x * divgr K B y.	Proof. by move=> Hx; rewrite /divgr remgrMl ?mulgA. Qed.	Lemma	divgrMl	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "divgr", "mulgA", "remgrMl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divgr_id K H x : x \in K -> divgr K H x = x.	Proof. by move=> Kx; rewrite /divgr remgr1 // invg1 mulg1. Qed.	Lemma	divgr_id	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "divgr", "invg1", "mulg1", "remgr1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_remgr K B x : x \in K * B -> remgr K B x \in B.	Proof. by rewrite -remgrP => /setIP[]. Qed.	Lemma	mem_remgr	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "remgr", "remgrP", "setIP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_divgr K B x : x \in K * B -> divgr K B x \in K.	Proof. by rewrite -remgrP inE rcoset_sym mem_rcoset => /andP[]. Qed.	Lemma	mem_divgr	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "divgr", "inE", "mem_rcoset", "rcoset_sym", "remgrP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tiKH : K :&: H = 1.		Hypothesis	tiKH	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
remgr_id x : x \in H -> remgr K H x = x.	Proof. move=> Hx; apply/eqP; rewrite eq_mulgV1 (sameP eqP set1gP) -tiKH inE. rewrite -mem_rcoset groupMr ?groupV // -in_setI remgrP. by apply: subsetP Hx; apply: mulG_subr. Qed.	Lemma	remgr_id	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "eq_mulgV1", "groupMr", "groupV", "inE", "in_setI", "mem_rcoset", "mulG_subr", "remgr", "remgrP", "set1gP", "subsetP", "tiKH" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
remgrMid x y : x \in K -> y \in H -> remgr K H (x * y) = y.	Proof. by move=> Kx Hy; rewrite remgrMl ?remgr_id. Qed.	Lemma	remgrMid	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "remgr", "remgrMl", "remgr_id" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divgrMid x y : x \in K -> y \in H -> divgr K H (x * y) = x.	Proof. by move=> Kx Hy; rewrite /divgr remgrMid ?mulgK. Qed.	Lemma	divgrMid	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "divgr", "mulgK", "remgrMid" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subcent_TImulg K H A : K :&: H = 1 -> A \subset 'N(K) :&: 'N(H) -> 'C_K(A) * 'C_H(A) = 'C_(K * H)(A).	Proof. move=> tiKH /subsetIP[nKA nHA]; apply/eqP. rewrite group_modl ?subsetIr // eqEsubset setSI ?mulSg ?subsetIl //=. apply/subsetP=> _ /setIP[/mulsgP[x y Kx Hy ->] cAxy]. rewrite inE cAxy mem_mulg // inE Kx /=. apply/centP=> z Az; apply/commgP/conjg_fixP. move/commgP/conjg_fixP/(congr1 (divgr K H)): (centP cAxy z Az...	Lemma	subcent_TImulg	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "centP", "commgP", "conjMg", "conjg_fixP", "divgr", "divgrMid", "eqEsubset", "group_modl", "inE", "memJ_norm", "mem_mulg", "mulSg", "mulsgP", "setIP", "setSI", "subsetIP", "subsetIl", "subsetIr", "subsetP", "tiKH" ]	Intersection of a centraliser with a disjoint product.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
complP H A B : reflect (A :&: H = 1 /\ A * H = B) (H \in [complements to A in B]).	Proof. by apply: (iffP setIdP); case; split; apply/eqP. Qed.	Lemma	complP	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "setIdP", "split", "to" ]	Complements, and splitting.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
splitsP B A : reflect (exists H, H \in [complements to A in B]) [splits B, over A].	Proof. exact: set0Pn. Qed.	Lemma	splitsP	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "set0Pn", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
complgC H K G : (H \in [complements to K in G]) = (K \in [complements to H in G]).	Proof. rewrite !inE setIC; congr (_ && _). by apply/eqP/eqP=> defG; rewrite -(comm_group_setP _) // defG groupP. Qed.	Lemma	complgC	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "comm_group_setP", "defG", "groupP", "inE", "setIC", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
complH_K : H \in [complements to K in G].		Hypothesis	complH_K	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
remgrM : K <\| G -> {in G &, {morph remgr K H : x y / x * y}}.	Proof. case/normalP=> _; case/complP: complH_K => tiKH <- nK_KH x y KHx KHy. rewrite {1}(divgr_eq K H y) mulgA (conjgCV x) {2}(divgr_eq K H x) -mulgA. rewrite -[X in _ * X]mulgA mulgA remgrMid //; last first. by rewrite groupMl mem_remgr. by rewrite groupMl !(=^~ mem_conjg, nK_KH, mem_divgr). Qed.	Lemma	remgrM	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "complH_K", "complP", "conjgCV", "divgr_eq", "groupMl", "last", "mem_conjg", "mem_divgr", "mem_remgr", "mulgA", "normalP", "remgr", "remgrMid", "tiKH" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divgrM : H \subset 'C(K) -> {in G &, {morph divgr K H : x y / x * y}}.	Proof. move=> cKH; have /complP[_ defG] := complH_K. have nsKG: K <\| G by rewrite -defG -cent_joinEr // normalYl cents_norm. move=> x y Gx Gy; rewrite {1}/divgr remgrM // invMg -!mulgA (mulgA y). by congr (_ * _); rewrite -(centsP cKH) ?groupV ?(mem_remgr, mem_divgr, defG). Qed.	Lemma	divgrM	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "cKH", "cent_joinEr", "centsP", "cents_norm", "complH_K", "complP", "defG", "divgr", "groupV", "invMg", "mem_divgr", "mem_remgr", "mulgA", "normalYl", "nsKG", "remgrM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod1g : left_id 1 sdprod.	Proof. by move=> A; rewrite /sdprod subsetIl pprod1g. Qed.	Lemma	sdprod1g	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "pprod1g", "sdprod", "subsetIl" ]	Semi-direct product	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprodg1 : right_id 1 sdprod.	Proof. by move=> A; rewrite /sdprod subsetIr pprodg1. Qed.	Lemma	sdprodg1	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "pprodg1", "sdprod", "subsetIr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprodP A B G : A ><\| B = G -> [/\ are_groups A B, A * B = G, B \subset 'N(A) & A :&: B = 1].	Proof. rewrite /sdprod; case: ifP => [trAB \| _ /group_not0[] //]. case/pprodP=> gAB defG nBA; split=> {defG nBA}//. by case: gAB trAB => H K -> -> /trivgP. Qed.	Lemma	sdprodP	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "are_groups", "defG", "group_not0", "nBA", "pprodP", "sdprod", "split", "trivgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprodE K H : H \subset 'N(K) -> K :&: H = 1 -> K ><\| H = K * H.	Proof. by move=> nKH tiKH; rewrite /sdprod tiKH subxx pprodE. Qed.	Lemma	sdprodE	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "nKH", "pprodE", "sdprod", "subxx", "tiKH" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprodEY K H : H \subset 'N(K) -> K :&: H = 1 -> K ><\| H = K <*> H.	Proof. by move=> nKH tiKH; rewrite sdprodE ?norm_joinEr. Qed.	Lemma	sdprodEY	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "nKH", "norm_joinEr", "sdprodE", "tiKH" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprodWpp A B G : A ><\| B = G -> pprod A B = G.	Proof. by case/sdprodP=> [[K H -> ->] <- /pprodE]. Qed.	Lemma	sdprodWpp	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "pprod", "pprodE", "sdprodP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprodW A B G : A ><\| B = G -> A * B = G.	Proof. by move/sdprodWpp/pprodW. Qed.	Lemma	sdprodW	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "pprodW", "sdprodWpp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprodWC A B G : A ><\| B = G -> B * A = G.	Proof. by move/sdprodWpp/pprodWC. Qed.	Lemma	sdprodWC	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "pprodWC", "sdprodWpp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprodWY A B G : A ><\| B = G -> A <*> B = G.	Proof. by move/sdprodWpp/pprodWY. Qed.	Lemma	sdprodWY	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "pprodWY", "sdprodWpp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprodJ A B x : (A ><\| B) :^ x = A :^ x ><\| B :^ x.	Proof. rewrite /sdprod -conjIg sub_conjg conjs1g -pprodJ. by case: ifP => _ //; apply: imset0. Qed.	Lemma	sdprodJ	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "conjIg", "conjs1g", "imset0", "pprodJ", "sdprod", "sub_conjg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod_context G K H : K ><\| H = G -> [/\ K <\| G, H \subset G, K * H = G, H \subset 'N(K) & K :&: H = 1].	Proof. case/sdprodP=> _ <- nKH tiKH. by rewrite /normal mulG_subl mulG_subr mulG_subG normG. Qed.	Lemma	sdprod_context	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "mulG_subG", "mulG_subl", "mulG_subr", "nKH", "normG", "normal", "sdprodP", "tiKH" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod_compl G K H : K ><\| H = G -> H \in [complements to K in G].	Proof. by case/sdprodP=> _ mulKH _ tiKH; apply/complP. Qed.	Lemma	sdprod_compl	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "complP", "sdprodP", "tiKH", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod_normal_complP G K H : K <\| G -> reflect (K ><\| H = G) (K \in [complements to H in G]).	Proof. case/andP=> _ nKG; rewrite complgC. apply: (iffP idP); [case/complP=> tiKH mulKH \| exact: sdprod_compl]. by rewrite sdprodE ?(subset_trans _ nKG) // -mulKH mulG_subr. Qed.	Lemma	sdprod_normal_complP	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "complP", "complgC", "mulG_subr", "nKG", "sdprodE", "sdprod_compl", "subset_trans", "tiKH", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod_card G A B : A ><\| B = G -> (#\|A\| * #\|B\|)%N = #\|G\|.	Proof. by case/sdprodP=> [[H K -> ->] <- _ /TI_cardMg]. Qed.	Lemma	sdprod_card	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "TI_cardMg", "sdprodP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod_isom G A B : A ><\| B = G -> {nAB : B \subset 'N(A) \| isom B (G / A) (restrm nAB (coset A))}.	Proof. case/sdprodP=> [[K H -> ->] <- nKH tiKH]. by exists nKH; rewrite quotientMidl quotient_isom. Qed.	Lemma	sdprod_isom	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "coset", "isom", "nKH", "quotientMidl", "quotient_isom", "restrm", "sdprodP", "tiKH" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod_isog G A B : A ><\| B = G -> B \isog G / A.	Proof. by case/sdprod_isom=> nAB; apply: isom_isog. Qed.	Lemma	sdprod_isog	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "isog", "isom_isog", "sdprod_isom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod_subr G A B M : A ><\| B = G -> M \subset B -> A ><\| M = A <*> M.	Proof. case/sdprodP=> [[K H -> ->] _ nKH tiKH] sMH. by rewrite sdprodEY ?(subset_trans sMH) //; apply/trivgP; rewrite -tiKH setIS. Qed.	Lemma	sdprod_subr	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "nKH", "sdprodEY", "sdprodP", "setIS", "subset_trans", "tiKH", "trivgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
index_sdprod G A B : A ><\| B = G -> #\|B\| = #\|G : A\|.	Proof. case/sdprodP=> [[K H -> ->] <- _ tiHK]. by rewrite indexMg -indexgI setIC tiHK indexg1. Qed.	Lemma	index_sdprod	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "indexMg", "indexg1", "indexgI", "sdprodP", "setIC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
index_sdprodr G A B M : A ><\| B = G -> M \subset B -> #\|B : M\| = #\|G : A <*> M\|.	Proof. move=> defG; case/sdprodP: defG (defG) => [[K H -> ->] mulKH nKH _] defG sMH. rewrite -!divgS //=; first by rewrite -genM_join gen_subG -mulKH mulgS. by rewrite -(sdprod_card defG) -(sdprod_card (sdprod_subr defG sMH)) divnMl. Qed.	Lemma	index_sdprodr	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "defG", "divgS", "divnMl", "genM_join", "gen_subG", "mulgS", "nKH", "sdprodP", "sdprod_card", "sdprod_subr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_sdprodr_isom G A B M : A ><\| B = G -> M <\| B -> {f : {morphism B / M >-> coset_of (A <> M)} \| isom (B / M) (G / (A <> M)) f & forall L, L \subset B -> f @* (L / M) = A <> L / (A <> M)}.	Proof. move=> defG nsMH; have [defA defB]: A = <<A>>%G /\ B = <<B>>%G. by have [[K1 H1 -> ->] _ _ _] := sdprodP defG; rewrite /= !genGid. do [rewrite {}defA {}defB; move: {A}<<A>>%G {B}<<B>>%G => K H] in defG nsMH *. have [[nKH /isomP[injKH imKH]] sMH] := (sdprod_isom defG, normal_sub nsMH). have [[nsKG sHG mulKH _ _...	Lemma	quotient_sdprodr_isom	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "coset_of", "cosetpre_normal", "defG", "dom", "domP", "genGid", "inj_f", "injmK", "injm_comp", "injm_quotm", "isom", "isomP", "joing1G", "joing_subl", "last", "morphim_comp", "morphim_quotm", "morphism", "morphpre_quotm", "nKH", "normal_norm", "normal_sub", "no...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_sdprodr_isog G A B M : A ><\| B = G -> M <\| B -> B / M \isog G / (A <*> M).	Proof. move=> defG; case/sdprodP: defG (defG) => [[K H -> ->] _ _ _] => defG nsMH. by have [h /isom_isog->] := quotient_sdprodr_isom defG nsMH. Qed.	Lemma	quotient_sdprodr_isog	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "defG", "isog", "isom_isog", "quotient_sdprodr_isom", "sdprodP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod_modl A B G H : A ><\| B = G -> A \subset H -> A ><\| (B :&: H) = G :&: H.	Proof. case/sdprodP=> {A B} [[A B -> ->]] <- nAB tiAB sAH. rewrite -group_modl ?sdprodE ?subIset ?nAB //. by rewrite setIA tiAB (setIidPl _) ?sub1G. Qed.	Lemma	sdprod_modl	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "group_modl", "sdprodE", "sdprodP", "setIA", "setIidPl", "sub1G", "subIset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod_modr A B G H : A ><\| B = G -> B \subset H -> (H :&: A) ><\| B = H :&: G.	Proof. case/sdprodP=> {A B}[[A B -> ->]] <- nAB tiAB sAH. rewrite -group_modr ?sdprodE ?normsI // ?normsG //. by rewrite -setIA tiAB (setIidPr _) ?sub1G. Qed.	Lemma	sdprod_modr	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "group_modr", "normsG", "normsI", "sdprodE", "sdprodP", "setIA", "setIidPr", "sub1G" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subcent_sdprod B C G A : B ><\| C = G -> A \subset 'N(B) :&: 'N(C) -> 'C_B(A) ><\| 'C_C(A) = 'C_G(A).	Proof. case/sdprodP=> [[H K -> ->] <- nHK tiHK] nHKA {B C G}. rewrite sdprodE ?subcent_TImulg ?normsIG //. by rewrite -setIIl tiHK (setIidPl (sub1G _)). Qed.	Lemma	subcent_sdprod	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "nHK", "normsIG", "sdprodE", "sdprodP", "setIIl", "setIidPl", "sub1G", "subcent_TImulg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod_recl n G K H K1 : #\|G\| <= n -> K ><\| H = G -> K1 \proper K -> H \subset 'N(K1) -> exists G1 : {group gT}, [/\ #\|G1\| < n, G1 \subset G & K1 ><\| H = G1].	Proof. move=> leGn; case/sdprodP=> _ defG nKH tiKH ltK1K nK1H. have tiK1H: K1 :&: H = 1 by apply/trivgP; rewrite -tiKH setSI ?proper_sub. exists (K1 <*> H)%G; rewrite /= -defG sdprodE // norm_joinEr //. rewrite ?mulSg ?proper_sub ?(leq_trans _ leGn) //=. by rewrite -defG ?TI_cardMg // ltn_pmul2r ?proper_card. Qed.	Lemma	sdprod_recl	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "G1", "TI_cardMg", "apply", "defG", "gT", "group", "leq_trans", "ltn_pmul2r", "mulSg", "nKH", "norm_joinEr", "proper", "proper_card", "proper_sub", "sdprodE", "sdprodP", "setSI", "tiKH", "trivgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod_recr n G K H H1 : #\|G\| <= n -> K ><\| H = G -> H1 \proper H -> exists G1 : {group gT}, [/\ #\|G1\| < n, G1 \subset G & K ><\| H1 = G1].	Proof. move=> leGn; case/sdprodP=> _ defG nKH tiKH ltH1H. have [sH1H _] := andP ltH1H; have nKH1 := subset_trans sH1H nKH. have tiKH1: K :&: H1 = 1 by apply/trivgP; rewrite -tiKH setIS. exists (K <*> H1)%G; rewrite /= -defG sdprodE // norm_joinEr //. rewrite ?mulgS // ?(leq_trans _ leGn) //=. by rewrite -defG ?TI_cardM...	Lemma	sdprod_recr	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "G1", "TI_cardMg", "apply", "defG", "gT", "group", "leq_trans", "ltn_pmul2l", "mulgS", "nKH", "norm_joinEr", "proper", "proper_card", "sdprodE", "sdprodP", "setIS", "subset_trans", "tiKH", "trivgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_sdprod G A B x : A ><\| B = G -> x \in G -> exists y, exists z, [/\ y \in A, z \in B, x = y * z & {in A & B, forall u t, x = u * t -> u = y /\ t = z}].	Proof. case/sdprodP=> [[K H -> ->{A B}] <- _ tiKH] /mulsgP[y z Ky Hz ->{x}]. exists y; exists z; split=> // u t Ku Ht eqyzut. move: (congr1 (divgr K H) eqyzut) (congr1 (remgr K H) eqyzut). by rewrite !remgrMid // !divgrMid. Qed.	Lemma	mem_sdprod	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "divgr", "divgrMid", "mulsgP", "remgr", "remgrMid", "sdprodP", "split", "tiKH" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cprod1g : left_id 1 cprod.	Proof. by move=> A; rewrite /cprod cents1 pprod1g. Qed.	Lemma	cprod1g	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "cents1", "cprod", "pprod1g" ]	Central product	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cprodg1 : right_id 1 cprod.	Proof. by move=> A; rewrite /cprod sub1G pprodg1. Qed.	Lemma	cprodg1	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "cprod", "pprodg1", "sub1G" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cprodP A B G : A \* B = G -> [/\ are_groups A B, A * B = G & B \subset 'C(A)].	Proof. by rewrite /cprod; case: ifP => [cAB /pprodP[] \| _ /group_not0[]]. Qed.	Lemma	cprodP	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "are_groups", "cprod", "group_not0", "pprodP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

ex_maxgroup : (exists G, gP G) -> {G : {group gT} | maxgroup G gP}.

Proof. move=> exP; have [A maxA]: {A | maxgroup A gP}. apply: ex_maxset; case: exP => G gPG. by exists (G : {set gT}); rewrite groupP genGidG. by exists <<A>>%G; rewrite /= gen_set_id; case/andP: (maxsetp maxA). Qed.

Lemma

ex_maxgroup

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "exP", "ex_maxset", "gPG", "gT", "genGidG", "gen_set_id", "group", "groupP", "maxA", "maxgroup", "maxsetp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ex_mingroup : (exists G, gP G) -> {G : {group gT} | mingroup G gP}.

Proof. move=> exP; have [A minA]: {A | mingroup A gP}. apply: ex_minset; case: exP => G gPG. by exists (G : {set gT}); rewrite groupP genGidG. by exists <<A>>%G; rewrite /= gen_set_id; case/andP: (minsetp minA). Qed.

Lemma

ex_mingroup

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "exP", "ex_minset", "gPG", "gT", "genGidG", "gen_set_id", "group", "groupP", "minA", "mingroup", "minsetp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mingroupP : reflect (gP G /\ forall H, gP H -> H \subset G -> H :=: G) (mingroup G gP).

Proof. apply: (iffP minsetP); rewrite /= groupP genGidG /= => [] [-> minG]. by split=> // H gPH sGH; apply: minG; rewrite // groupP genGidG. by split=> // A; case/andP=> gA gPA; rewrite -(gen_set_id gA); apply: minG. Qed.

Lemma

mingroupP

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "genGidG", "gen_set_id", "groupP", "mingroup", "minsetP", "sGH", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

maxgroupP : reflect (gP G /\ forall H, gP H -> G \subset H -> H :=: G) (maxgroup G gP).

Proof. apply: (iffP maxsetP); rewrite /= groupP genGidG /= => [] [-> maxG]. by split=> // H gPH sGH; apply: maxG; rewrite // groupP genGidG. by split=> // A; case/andP=> gA gPA; rewrite -(gen_set_id gA); apply: maxG. Qed.

Lemma

maxgroupP

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "genGidG", "gen_set_id", "groupP", "maxgroup", "maxsetP", "sGH", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

maxgroupp : maxgroup G gP -> gP G.

Proof. by case/maxgroupP. Qed.

Lemma

maxgroupp

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "maxgroup", "maxgroupP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mingroupp : mingroup G gP -> gP G.

Proof. by case/mingroupP. Qed.

Lemma

mingroupp

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "mingroup", "mingroupP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gPG : gP G.

Hypothesis

gPG

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

maxgroup_exists : {H : {group gT} | maxgroup H gP & G \subset H}.

Proof. have [A maxA sGA]: {A | maxgroup A gP & G \subset A}. by apply: maxset_exists; rewrite groupP genGidG. by exists <<A>>%G; rewrite /= gen_set_id; case/andP: (maxsetp maxA). Qed.

Lemma

maxgroup_exists

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "gT", "genGidG", "gen_set_id", "group", "groupP", "maxA", "maxgroup", "maxset_exists", "maxsetp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mingroup_exists : {H : {group gT} | mingroup H gP & H \subset G}.

Proof. have [A maxA sGA]: {A | mingroup A gP & A \subset G}. by apply: minset_exists; rewrite groupP genGidG. by exists <<A>>%G; rewrite /= gen_set_id; case/andP: (minsetp maxA). Qed.

Lemma

mingroup_exists

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "gT", "genGidG", "gen_set_id", "group", "groupP", "maxA", "mingroup", "minset_exists", "minsetp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'max' A 'of' G | gP ]"

:= (maxgroup A (fun G : {group _} => gP)) : group_scope.

Notation

[ 'max' A 'of' G | gP ]

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "group", "maxgroup" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'max' G | gP ]"

:= [max gval G of G | gP] : group_scope.

Notation

[ 'max' G | gP ]

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "max" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'max' A 'of' G | gP & gQ ]"

:= [max A of G | gP && gQ] : group_scope.

Notation

[ 'max' A 'of' G | gP & gQ ]

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "max" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'max' G | gP & gQ ]"

:= [max G | gP && gQ] : group_scope.

Notation

[ 'max' G | gP & gQ ]

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "max" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'min' A 'of' G | gP ]"

:= (mingroup A (fun G : {group _} => gP)) : group_scope.

Notation

[ 'min' A 'of' G | gP ]

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "group", "mingroup" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'min' G | gP ]"

:= [min gval G of G | gP] : group_scope.

Notation

[ 'min' G | gP ]

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "min" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'min' A 'of' G | gP & gQ ]"

:= [min A of G | gP && gQ] : group_scope.

Notation

[ 'min' A 'of' G | gP & gQ ]

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "min" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'min' G | gP & gQ ]"

:= [min G | gP && gQ] : group_scope.

Notation

[ 'min' G | gP & gQ ]

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "min" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

partial_product A B

:= if A == 1 then B else if B == 1 then A else if [&& group_set A, group_set B & B \subset 'N(A)] then A * B else set0.

Definition

partial_product

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "group_set", "set0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

semidirect_product A B

:= if A :&: B \subset 1%G then partial_product A B else set0.

Definition

semidirect_product

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "partial_product", "set0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

central_product A B

:= if B \subset 'C(A) then partial_product A B else set0.

Definition

central_product

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "partial_product", "set0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

direct_product A B

:= if A :&: B \subset 1%G then central_product A B else set0.

Definition

direct_product

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "central_product", "set0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

complements_to_in A B

:= [set K : {group gT} | A :&: K == 1 & A * K == B].

Definition

complements_to_in

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "gT", "group" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

splits_over B A

:= complements_to_in A B != set0.

Definition

splits_over

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "complements_to_in", "set0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

remgr A B x

:= repr (A :* x :&: B).

Definition

remgr

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "repr" ]

Product remainder functions -- right variant only.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divgr A B x

:= x * (remgr A B x)^-1.

Definition

divgr

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "remgr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pprod

:= (partial_product _).

Notation

pprod

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "partial_product" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod

:= (semidirect_product _).

Notation

sdprod

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "semidirect_product" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cprod

:= (central_product _).

Notation

cprod

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "central_product" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dprod

:= (direct_product _).

Notation

dprod

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "direct_product" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"G ><| H"

:= (sdprod G H)%g (at level 40, left associativity) : group_scope.

Notation

G ><| H

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "sdprod" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"G \* H"

:= (cprod G H)%g (at level 40, left associativity) : group_scope.

Notation

G \* H

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "cprod" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"G \x H"

:= (dprod G H)%g (at level 40, left associativity) : group_scope.

Notation

G \x H

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "dprod" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'complements' 'to' A 'in' B ]"

:= (complements_to_in A B) (format "[ 'complements' 'to' A 'in' B ]") : group_scope.

Notation

[ 'complements' 'to' A 'in' B ]

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "complements_to_in" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'splits' B , 'over' A ]"

:= (splits_over B A) (format "[ 'splits' B , 'over' A ]") : group_scope.

Notation

[ 'splits' B , 'over' A ]

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "splits_over" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pprod

:= (partial_product gT).

Notation

pprod

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "gT", "partial_product" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod

:= (semidirect_product gT) (only parsing).

Notation

sdprod

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "gT", "semidirect_product" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cprod

:= (central_product gT) (only parsing).

Notation

cprod

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "central_product", "gT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dprod

:= (direct_product gT) (only parsing).

Notation

dprod

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "direct_product", "gT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pprod1g : left_id 1 pprod.

Proof. by move=> A; rewrite /pprod eqxx. Qed.

Lemma

pprod1g

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "eqxx", "pprod" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pprodg1 : right_id 1 pprod.

Proof. by move=> A; rewrite /pprod eqxx; case: eqP. Qed.

Lemma

pprodg1

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "eqxx", "pprod" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

are_groups A B : Prop

:= AreGroups K H of A = K & B = H.

Variant

are_groups

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group_not0 G : set0 <> G.

Proof. by move/setP/(_ 1); rewrite inE group1. Qed.

Lemma

group_not0

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "group1", "inE", "set0", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulg0 : right_zero (@set0 gT) mul.

Proof. by move=> A; apply/setP=> x; rewrite inE; apply/imset2P=> [[y z]]; rewrite inE. Qed.

Lemma

mulg0

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "gT", "imset2P", "inE", "mul", "set0", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mul0g : left_zero (@set0 gT) mul.

Proof. by move=> A; apply/setP=> x; rewrite inE; apply/imset2P=> [[y z]]; rewrite inE. Qed.

Lemma

mul0g

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "gT", "imset2P", "inE", "mul", "set0", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pprodP A B G : pprod A B = G -> [/\ are_groups A B, A * B = G & B \subset 'N(A)].

Proof. have Gnot0 := @group_not0 G; rewrite /pprod; do 2?case: eqP => [-> ->| _]. - by rewrite mul1g norms1; split; first exists 1%G G. - by rewrite mulg1 sub1G; split; first exists G 1%G. by case: and3P => // [[gA gB ->]]; split; first exists (Group gA) (Group gB). Qed.

Lemma

pprodP

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "are_groups", "group_not0", "mul1g", "mulg1", "norms1", "pprod", "split", "sub1G" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pprodE K H : H \subset 'N(K) -> pprod K H = K * H.

Proof. move=> nKH; rewrite /pprod nKH !groupP /=. by do 2?case: eqP => [-> | _]; rewrite ?mulg1 ?mul1g. Qed.

Lemma

pprodE

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "groupP", "mul1g", "mulg1", "nKH", "pprod" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pprodEY K H : H \subset 'N(K) -> pprod K H = K <*> H.

Proof. by move=> nKH; rewrite pprodE ?norm_joinEr. Qed.

Lemma

pprodEY

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "nKH", "norm_joinEr", "pprod", "pprodE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pprodW A B G : pprod A B = G -> A * B = G.

Proof. by case/pprodP. Qed.

Lemma

pprodW

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "pprod", "pprodP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pprodWC A B G : pprod A B = G -> B * A = G.

Proof. by case/pprodP=> _ <- /normC. Qed.

Lemma

pprodWC

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "normC", "pprod", "pprodP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pprodWY A B G : pprod A B = G -> A <*> B = G.

Proof. by case/pprodP=> [[K H -> ->] <- /norm_joinEr]. Qed.

Lemma

pprodWY

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "norm_joinEr", "pprod", "pprodP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pprodJ A B x : pprod A B :^ x = pprod (A :^ x) (B :^ x).

Proof. rewrite /pprod !conjsg_eq1 !group_setJ normJ conjSg -conjsMg. by do 3?case: ifP => // _; apply: conj0g. Qed.

Lemma

pprodJ

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "conj0g", "conjSg", "conjsMg", "conjsg_eq1", "group_setJ", "normJ", "pprod" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

remgrMl K B x y : y \in K -> remgr K B (y * x) = remgr K B x.

Proof. by move=> Ky; rewrite {1}/remgr rcosetM rcoset_id. Qed.

Lemma

remgrMl

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "rcosetM", "rcoset_id", "remgr" ]

Properties of the remainders

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

remgrP K B x : (remgr K B x \in K :* x :&: B) = (x \in K * B).

Proof. set y := _ x; apply/idP/mulsgP=> [|[g b Kg Bb x_gb]]. rewrite inE rcoset_sym mem_rcoset => /andP[Kxy' By]. by exists (x * y^-1) y; rewrite ?mulgKV. by apply: (mem_repr b); rewrite inE rcoset_sym mem_rcoset x_gb mulgK Kg. Qed.

Lemma

remgrP

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "inE", "mem_rcoset", "mem_repr", "mulgK", "mulgKV", "mulsgP", "rcoset_sym", "remgr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

remgr1 K H x : x \in K -> remgr K H x = 1.

Proof. by move=> Kx; rewrite /remgr rcoset_id ?repr_group. Qed.

Lemma

remgr1

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "rcoset_id", "remgr", "repr_group" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divgr_eq A B x : x = divgr A B x * remgr A B x.

Proof. by rewrite mulgKV. Qed.

Lemma

divgr_eq

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "divgr", "mulgKV", "remgr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divgrMl K B x y : x \in K -> divgr K B (x * y) = x * divgr K B y.

Proof. by move=> Hx; rewrite /divgr remgrMl ?mulgA. Qed.

Lemma

divgrMl

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "divgr", "mulgA", "remgrMl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divgr_id K H x : x \in K -> divgr K H x = x.

Proof. by move=> Kx; rewrite /divgr remgr1 // invg1 mulg1. Qed.

Lemma

divgr_id

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "divgr", "invg1", "mulg1", "remgr1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_remgr K B x : x \in K * B -> remgr K B x \in B.

Proof. by rewrite -remgrP => /setIP[]. Qed.

Lemma

mem_remgr

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "remgr", "remgrP", "setIP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_divgr K B x : x \in K * B -> divgr K B x \in K.

Proof. by rewrite -remgrP inE rcoset_sym mem_rcoset => /andP[]. Qed.

Lemma

mem_divgr

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "divgr", "inE", "mem_rcoset", "rcoset_sym", "remgrP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tiKH : K :&: H = 1.

Hypothesis

tiKH

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

remgr_id x : x \in H -> remgr K H x = x.

Proof. move=> Hx; apply/eqP; rewrite eq_mulgV1 (sameP eqP set1gP) -tiKH inE. rewrite -mem_rcoset groupMr ?groupV // -in_setI remgrP. by apply: subsetP Hx; apply: mulG_subr. Qed.

Lemma

remgr_id

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "eq_mulgV1", "groupMr", "groupV", "inE", "in_setI", "mem_rcoset", "mulG_subr", "remgr", "remgrP", "set1gP", "subsetP", "tiKH" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

remgrMid x y : x \in K -> y \in H -> remgr K H (x * y) = y.

Proof. by move=> Kx Hy; rewrite remgrMl ?remgr_id. Qed.

Lemma

remgrMid

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "remgr", "remgrMl", "remgr_id" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divgrMid x y : x \in K -> y \in H -> divgr K H (x * y) = x.

Proof. by move=> Kx Hy; rewrite /divgr remgrMid ?mulgK. Qed.

Lemma

divgrMid

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "divgr", "mulgK", "remgrMid" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subcent_TImulg K H A : K :&: H = 1 -> A \subset 'N(K) :&: 'N(H) -> 'C_K(A) * 'C_H(A) = 'C_(K * H)(A).

Proof. move=> tiKH /subsetIP[nKA nHA]; apply/eqP. rewrite group_modl ?subsetIr // eqEsubset setSI ?mulSg ?subsetIl //=. apply/subsetP=> _ /setIP[/mulsgP[x y Kx Hy ->] cAxy]. rewrite inE cAxy mem_mulg // inE Kx /=. apply/centP=> z Az; apply/commgP/conjg_fixP. move/commgP/conjg_fixP/(congr1 (divgr K H)): (centP cAxy z Az...

Lemma

subcent_TImulg

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "centP", "commgP", "conjMg", "conjg_fixP", "divgr", "divgrMid", "eqEsubset", "group_modl", "inE", "memJ_norm", "mem_mulg", "mulSg", "mulsgP", "setIP", "setSI", "subsetIP", "subsetIl", "subsetIr", "subsetP", "tiKH" ]

Intersection of a centraliser with a disjoint product.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

complP H A B : reflect (A :&: H = 1 /\ A * H = B) (H \in [complements to A in B]).

Proof. by apply: (iffP setIdP); case; split; apply/eqP. Qed.

Lemma

complP

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "setIdP", "split", "to" ]

Complements, and splitting.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

splitsP B A : reflect (exists H, H \in [complements to A in B]) [splits B, over A].

Proof. exact: set0Pn. Qed.

Lemma

splitsP

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "set0Pn", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

complgC H K G : (H \in [complements to K in G]) = (K \in [complements to H in G]).

Proof. rewrite !inE setIC; congr (_ && _). by apply/eqP/eqP=> defG; rewrite -(comm_group_setP _) // defG groupP. Qed.

Lemma

complgC

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "comm_group_setP", "defG", "groupP", "inE", "setIC", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

complH_K : H \in [complements to K in G].

Hypothesis

complH_K

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

remgrM : K <| G -> {in G &, {morph remgr K H : x y / x * y}}.

Proof. case/normalP=> _; case/complP: complH_K => tiKH <- nK_KH x y KHx KHy. rewrite {1}(divgr_eq K H y) mulgA (conjgCV x) {2}(divgr_eq K H x) -mulgA. rewrite -[X in _ * X]mulgA mulgA remgrMid //; last first. by rewrite groupMl mem_remgr. by rewrite groupMl !(=^~ mem_conjg, nK_KH, mem_divgr). Qed.

Lemma

remgrM

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "complH_K", "complP", "conjgCV", "divgr_eq", "groupMl", "last", "mem_conjg", "mem_divgr", "mem_remgr", "mulgA", "normalP", "remgr", "remgrMid", "tiKH" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divgrM : H \subset 'C(K) -> {in G &, {morph divgr K H : x y / x * y}}.

Proof. move=> cKH; have /complP[_ defG] := complH_K. have nsKG: K <| G by rewrite -defG -cent_joinEr // normalYl cents_norm. move=> x y Gx Gy; rewrite {1}/divgr remgrM // invMg -!mulgA (mulgA y). by congr (_ * _); rewrite -(centsP cKH) ?groupV ?(mem_remgr, mem_divgr, defG). Qed.

Lemma

divgrM

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "cKH", "cent_joinEr", "centsP", "cents_norm", "complH_K", "complP", "defG", "divgr", "groupV", "invMg", "mem_divgr", "mem_remgr", "mulgA", "normalYl", "nsKG", "remgrM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod1g : left_id 1 sdprod.

Proof. by move=> A; rewrite /sdprod subsetIl pprod1g. Qed.

Lemma

sdprod1g

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "pprod1g", "sdprod", "subsetIl" ]

Semi-direct product

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprodg1 : right_id 1 sdprod.

Proof. by move=> A; rewrite /sdprod subsetIr pprodg1. Qed.

Lemma

sdprodg1

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "pprodg1", "sdprod", "subsetIr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprodP A B G : A ><| B = G -> [/\ are_groups A B, A * B = G, B \subset 'N(A) & A :&: B = 1].

Proof. rewrite /sdprod; case: ifP => [trAB | _ /group_not0[] //]. case/pprodP=> gAB defG nBA; split=> {defG nBA}//. by case: gAB trAB => H K -> -> /trivgP. Qed.

Lemma

sdprodP

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "are_groups", "defG", "group_not0", "nBA", "pprodP", "sdprod", "split", "trivgP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprodE K H : H \subset 'N(K) -> K :&: H = 1 -> K ><| H = K * H.

Proof. by move=> nKH tiKH; rewrite /sdprod tiKH subxx pprodE. Qed.

Lemma

sdprodE

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "nKH", "pprodE", "sdprod", "subxx", "tiKH" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprodEY K H : H \subset 'N(K) -> K :&: H = 1 -> K ><| H = K <*> H.

Proof. by move=> nKH tiKH; rewrite sdprodE ?norm_joinEr. Qed.

Lemma

sdprodEY

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "nKH", "norm_joinEr", "sdprodE", "tiKH" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprodWpp A B G : A ><| B = G -> pprod A B = G.

Proof. by case/sdprodP=> [[K H -> ->] <- /pprodE]. Qed.

Lemma

sdprodWpp

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "pprod", "pprodE", "sdprodP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprodW A B G : A ><| B = G -> A * B = G.

Proof. by move/sdprodWpp/pprodW. Qed.

Lemma

sdprodW

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "pprodW", "sdprodWpp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprodWC A B G : A ><| B = G -> B * A = G.

Proof. by move/sdprodWpp/pprodWC. Qed.

Lemma

sdprodWC

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "pprodWC", "sdprodWpp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprodWY A B G : A ><| B = G -> A <*> B = G.

Proof. by move/sdprodWpp/pprodWY. Qed.

Lemma

sdprodWY

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "pprodWY", "sdprodWpp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprodJ A B x : (A ><| B) :^ x = A :^ x ><| B :^ x.

Proof. rewrite /sdprod -conjIg sub_conjg conjs1g -pprodJ. by case: ifP => _ //; apply: imset0. Qed.

Lemma

sdprodJ

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "conjIg", "conjs1g", "imset0", "pprodJ", "sdprod", "sub_conjg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod_context G K H : K ><| H = G -> [/\ K <| G, H \subset G, K * H = G, H \subset 'N(K) & K :&: H = 1].

Proof. case/sdprodP=> _ <- nKH tiKH. by rewrite /normal mulG_subl mulG_subr mulG_subG normG. Qed.

Lemma

sdprod_context

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "mulG_subG", "mulG_subl", "mulG_subr", "nKH", "normG", "normal", "sdprodP", "tiKH" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod_compl G K H : K ><| H = G -> H \in [complements to K in G].

Proof. by case/sdprodP=> _ mulKH _ tiKH; apply/complP. Qed.

Lemma

sdprod_compl

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "complP", "sdprodP", "tiKH", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod_normal_complP G K H : K <| G -> reflect (K ><| H = G) (K \in [complements to H in G]).

Proof. case/andP=> _ nKG; rewrite complgC. apply: (iffP idP); [case/complP=> tiKH mulKH | exact: sdprod_compl]. by rewrite sdprodE ?(subset_trans _ nKG) // -mulKH mulG_subr. Qed.

Lemma

sdprod_normal_complP

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "complP", "complgC", "mulG_subr", "nKG", "sdprodE", "sdprod_compl", "subset_trans", "tiKH", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod_card G A B : A ><| B = G -> (#|A| * #|B|)%N = #|G|.

Proof. by case/sdprodP=> [[H K -> ->] <- _ /TI_cardMg]. Qed.

Lemma

sdprod_card

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "TI_cardMg", "sdprodP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod_isom G A B : A ><| B = G -> {nAB : B \subset 'N(A) | isom B (G / A) (restrm nAB (coset A))}.

Proof. case/sdprodP=> [[K H -> ->] <- nKH tiKH]. by exists nKH; rewrite quotientMidl quotient_isom. Qed.

Lemma

sdprod_isom

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "coset", "isom", "nKH", "quotientMidl", "quotient_isom", "restrm", "sdprodP", "tiKH" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod_isog G A B : A ><| B = G -> B \isog G / A.

Proof. by case/sdprod_isom=> nAB; apply: isom_isog. Qed.

Lemma

sdprod_isog

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "isog", "isom_isog", "sdprod_isom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod_subr G A B M : A ><| B = G -> M \subset B -> A ><| M = A <*> M.

Proof. case/sdprodP=> [[K H -> ->] _ nKH tiKH] sMH. by rewrite sdprodEY ?(subset_trans sMH) //; apply/trivgP; rewrite -tiKH setIS. Qed.

Lemma

sdprod_subr

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "nKH", "sdprodEY", "sdprodP", "setIS", "subset_trans", "tiKH", "trivgP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

index_sdprod G A B : A ><| B = G -> #|B| = #|G : A|.

Proof. case/sdprodP=> [[K H -> ->] <- _ tiHK]. by rewrite indexMg -indexgI setIC tiHK indexg1. Qed.

Lemma

index_sdprod

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "indexMg", "indexg1", "indexgI", "sdprodP", "setIC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

index_sdprodr G A B M : A ><| B = G -> M \subset B -> #|B : M| = #|G : A <*> M|.

Proof. move=> defG; case/sdprodP: defG (defG) => [[K H -> ->] mulKH nKH _] defG sMH. rewrite -!divgS //=; first by rewrite -genM_join gen_subG -mulKH mulgS. by rewrite -(sdprod_card defG) -(sdprod_card (sdprod_subr defG sMH)) divnMl. Qed.

Lemma

index_sdprodr

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "defG", "divgS", "divnMl", "genM_join", "gen_subG", "mulgS", "nKH", "sdprodP", "sdprod_card", "sdprod_subr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_sdprodr_isom G A B M : A ><| B = G -> M <| B -> {f : {morphism B / M >-> coset_of (A <*> M)} | isom (B / M) (G / (A <*> M)) f & forall L, L \subset B -> f @* (L / M) = A <*> L / (A <*> M)}.

Proof. move=> defG nsMH; have [defA defB]: A = <<A>>%G /\ B = <<B>>%G. by have [[K1 H1 -> ->] _ _ _] := sdprodP defG; rewrite /= !genGid. do [rewrite {}defA {}defB; move: {A}<<A>>%G {B}<<B>>%G => K H] in defG nsMH *. have [[nKH /isomP[injKH imKH]] sMH] := (sdprod_isom defG, normal_sub nsMH). have [[nsKG sHG mulKH _ _...

Lemma

quotient_sdprodr_isom

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "coset_of", "cosetpre_normal", "defG", "dom", "domP", "genGid", "inj_f", "injmK", "injm_comp", "injm_quotm", "isom", "isomP", "joing1G", "joing_subl", "last", "morphim_comp", "morphim_quotm", "morphism", "morphpre_quotm", "nKH", "normal_norm", "normal_sub", "no...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_sdprodr_isog G A B M : A ><| B = G -> M <| B -> B / M \isog G / (A <*> M).

Proof. move=> defG; case/sdprodP: defG (defG) => [[K H -> ->] _ _ _] => defG nsMH. by have [h /isom_isog->] := quotient_sdprodr_isom defG nsMH. Qed.

Lemma

quotient_sdprodr_isog

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "defG", "isog", "isom_isog", "quotient_sdprodr_isom", "sdprodP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod_modl A B G H : A ><| B = G -> A \subset H -> A ><| (B :&: H) = G :&: H.

Proof. case/sdprodP=> {A B} [[A B -> ->]] <- nAB tiAB sAH. rewrite -group_modl ?sdprodE ?subIset ?nAB //. by rewrite setIA tiAB (setIidPl _) ?sub1G. Qed.

Lemma

sdprod_modl

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "group_modl", "sdprodE", "sdprodP", "setIA", "setIidPl", "sub1G", "subIset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod_modr A B G H : A ><| B = G -> B \subset H -> (H :&: A) ><| B = H :&: G.

Proof. case/sdprodP=> {A B}[[A B -> ->]] <- nAB tiAB sAH. rewrite -group_modr ?sdprodE ?normsI // ?normsG //. by rewrite -setIA tiAB (setIidPr _) ?sub1G. Qed.

Lemma

sdprod_modr

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "group_modr", "normsG", "normsI", "sdprodE", "sdprodP", "setIA", "setIidPr", "sub1G" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subcent_sdprod B C G A : B ><| C = G -> A \subset 'N(B) :&: 'N(C) -> 'C_B(A) ><| 'C_C(A) = 'C_G(A).

Proof. case/sdprodP=> [[H K -> ->] <- nHK tiHK] nHKA {B C G}. rewrite sdprodE ?subcent_TImulg ?normsIG //. by rewrite -setIIl tiHK (setIidPl (sub1G _)). Qed.

Lemma

subcent_sdprod

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "nHK", "normsIG", "sdprodE", "sdprodP", "setIIl", "setIidPl", "sub1G", "subcent_TImulg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod_recl n G K H K1 : #|G| <= n -> K ><| H = G -> K1 \proper K -> H \subset 'N(K1) -> exists G1 : {group gT}, [/\ #|G1| < n, G1 \subset G & K1 ><| H = G1].

Proof. move=> leGn; case/sdprodP=> _ defG nKH tiKH ltK1K nK1H. have tiK1H: K1 :&: H = 1 by apply/trivgP; rewrite -tiKH setSI ?proper_sub. exists (K1 <*> H)%G; rewrite /= -defG sdprodE // norm_joinEr //. rewrite ?mulSg ?proper_sub ?(leq_trans _ leGn) //=. by rewrite -defG ?TI_cardMg // ltn_pmul2r ?proper_card. Qed.

Lemma

sdprod_recl

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "G1", "TI_cardMg", "apply", "defG", "gT", "group", "leq_trans", "ltn_pmul2r", "mulSg", "nKH", "norm_joinEr", "proper", "proper_card", "proper_sub", "sdprodE", "sdprodP", "setSI", "tiKH", "trivgP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod_recr n G K H H1 : #|G| <= n -> K ><| H = G -> H1 \proper H -> exists G1 : {group gT}, [/\ #|G1| < n, G1 \subset G & K ><| H1 = G1].

Proof. move=> leGn; case/sdprodP=> _ defG nKH tiKH ltH1H. have [sH1H _] := andP ltH1H; have nKH1 := subset_trans sH1H nKH. have tiKH1: K :&: H1 = 1 by apply/trivgP; rewrite -tiKH setIS. exists (K <*> H1)%G; rewrite /= -defG sdprodE // norm_joinEr //. rewrite ?mulgS // ?(leq_trans _ leGn) //=. by rewrite -defG ?TI_cardM...

Lemma

sdprod_recr

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "G1", "TI_cardMg", "apply", "defG", "gT", "group", "leq_trans", "ltn_pmul2l", "mulgS", "nKH", "norm_joinEr", "proper", "proper_card", "sdprodE", "sdprodP", "setIS", "subset_trans", "tiKH", "trivgP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_sdprod G A B x : A ><| B = G -> x \in G -> exists y, exists z, [/\ y \in A, z \in B, x = y * z & {in A & B, forall u t, x = u * t -> u = y /\ t = z}].

Proof. case/sdprodP=> [[K H -> ->{A B}] <- _ tiKH] /mulsgP[y z Ky Hz ->{x}]. exists y; exists z; split=> // u t Ku Ht eqyzut. move: (congr1 (divgr K H) eqyzut) (congr1 (remgr K H) eqyzut). by rewrite !remgrMid // !divgrMid. Qed.

Lemma

mem_sdprod

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "divgr", "divgrMid", "mulsgP", "remgr", "remgrMid", "sdprodP", "split", "tiKH" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cprod1g : left_id 1 cprod.

Proof. by move=> A; rewrite /cprod cents1 pprod1g. Qed.

Lemma

cprod1g

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "cents1", "cprod", "pprod1g" ]

Central product

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cprodg1 : right_id 1 cprod.

Proof. by move=> A; rewrite /cprod sub1G pprodg1. Qed.

Lemma

cprodg1

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "cprod", "pprodg1", "sub1G" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cprodP A B G : A \* B = G -> [/\ are_groups A B, A * B = G & B \subset 'C(A)].

Proof. by rewrite /cprod; case: ifP => [cAB /pprodP[] | _ /group_not0[]]. Qed.

Lemma

cprodP

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "are_groups", "cprod", "group_not0", "pprodP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d