Quelle permut.gd Sprache: unbekannt

#############################################################################
##
#W permut.gd Permutability GAP library ABB&ECL&RER
##
##
#Y Copyright (C) 2000-2018 Adolfo Ballester-Bolinches, Enric Cosme-Ll\'opez
#Y and Ramon Esteban-Romero
##
## This file contains declarations of methods for permutability
##

#############################################################################
##
#O ArePermutableSubgroups( <G>, , <V> ) . . . . for finite groups
#O ArePermutableSubgroups( , <V> )
##
## Returns true if and <V> permute, false otherwise.
##

ArePermutableSubgroups:=NewOperation("ArePermutableSubgroups",
 [IsGroup,IsGroup,IsGroup]);
#DeclareOperation("ArePermutableSubgroups", [IsGroup, IsGroup]);

#############################################################################
##
#O Permutizer( <G>, ) . . . . for finite groups
##
## Computes the permutizer of a subgroup U in G. This is the
## subgroup generated by all the cyclic subgroups of G which permute with U.
##
InParentFOA("Permutizer", IsGroup, IsGroup, DeclareAttribute);

DeclareSynonym("Permutiser", Permutizer);
DeclareSynonym("PermutiserOp", PermutizerOp);
DeclareSynonymAttr("PermutiserInParent", PermutizerInParent);

#############################################################################
##
#A AllSubnormalSubgroups( <G> ) . . . . for a finite group
##
## The default method constructs the lattice of subnormal subgroups.
##
DeclareAttribute("AllSubnormalSubgroups", IsGroup and IsFinite);

#############################################################################
##
#O IsPermutable( <G>, <H> ) . . . . for groups
##
## This operation returns true if H is permutable in G, false otherwise.
## H is permutable in G when H permutes with all subgroups of G.
##

InParentFOA("IsPermutable", IsGroup, IsGroup, DeclareProperty);

#############################################################################
##
#O IsFPermutable( <G>, <H>, <F> ) . . . . for groups
##
## This operation returns true if a subgroup H of G permutes
## with all the members of F, where F is a list of subgroups of G.
##

DeclareGlobalFunction( "IsFPermutable" );

#############################################################################
##
#O OneFSubgroupNotPermutingWith( <G>, <H>, <F> ) . . . . for groups
##
## This operation returns fail if H permutes with all members of F
## (a list of subgroups of G); otherwise it returns a member of F which
## does not permute with H.
##

DeclareGlobalFunction( "OneFSubgroupNotPermutingWith" );

#############################################################################
##
#O IsConjugatePermutable( <G>, <H> ) . . . . for groups
##
## This operation returns true if the subgroup H of G is
## conjugate-permutable, and false otherwise.
## H is conjugate-permutable in G when H permutes with H^g for every g of G.
##

InParentFOA("IsConjugatePermutable", IsGroup, IsGroup, DeclareProperty);

#############################################################################
##
#O IsSPermutable( <G>, <H> ) . . . . for groups
##
## This operation returns true if a subgroup H of G is S-permutable,
## and false otherwise.
## H is S-permutable in G when H permutes with all Sylow subgroups of G.
##

InParentFOA("IsSPermutable", IsGroup, IsGroup, DeclareProperty);

#############################################################################
##
#O IsSNPermutable( <G>, <H> ) . . . . for groups
##
## This operation returns true a subgroup H of a soluble group G
## is SN-permutable, and false otherwise.
## H is SN-permutable in G when H permutes with all system normalisers of G.
##

InParentFOA("IsSNPermutable", IsGroup, IsGroup, DeclareProperty);

#############################################################################
##
#O OneSubgroupNotPermutingWith( <G>, <H> ) . . . . for groups
##
## This operation finds a subgroup of G which does not permute with H
## if such a subgroup exists, otherwise it returns fail.
##
##
InParentFOA("OneSubgroupNotPermutingWith", IsGroup, IsGroup, DeclareAttribute);

#############################################################################
##
#A PrimesDividingSize( <G> ) . . . . for groups
##
## This attribute gives a list of primes dividing the size of G.
##
DeclareAttribute("PrimesDividingSize", IsGroup and IsFinite);

#############################################################################
##
#A SylowSubgroups( <G> ) . . . . for groups
##
## This attribute gives a list of one Sylow subgroup for every prime
## dividing the size of G
##

DeclareAttribute("SylowSubgroups", IsGroup and IsFinite);

#############################################################################
##
#O OneSylowSubgroupNotPermutingWith( <G>, <H> ) . . . . for groups
##
## This operation finds a Sylow subgroup of G which does not permute
## with H if such a subgroup exists, otherwise it returns fail.
##
##
InParentFOA("OneSylowSubgroupNotPermutingWith", IsGroup, IsGroup, DeclareAttribute);

#############################################################################
##
#O OneSystemNormalizerNotPermutingWith( <G>, <H> ) . . . . for groups
##
## This operation finds a system normaliser of the soluble group G which does
## not permute with H if such a subgroup exists, otherwise it returns fail.
##
##
InParentFOA("OneSystemNormalizerNotPermutingWith", IsGroup, IsGroup, DeclareAttribute);

Transatlantic(OneSystemNormalizerNotPermutingWith);
Transatlantic(OneSystemNormalizerNotPermutingWithOp);
Transatlantic(OneSystemNormalizerNotPermutingWithInParent);

#############################################################################
##
#O OneConjugateSubgroupNotPermutingWith( <G>, <H> ) . . . . for groups
##
## This operation finds a conjugate subgroup of H which does not permute
## with H if such a subgroup exists, otherwise it returns fail.
##

InParentFOA("OneConjugateSubgroupNotPermutingWith", IsGroup, IsGroup, DeclareAttribute);

#############################################################################
##
#O OneElementShowingNotWeaklyNormal( <G>, <H> ) . . . . for groups
##
## This operation finds an element g of G such that H is normal in <H, H^g>
## but H not normal in <H,g> if such an element exists, fail otherwise.
## H is weakly normal in G when H normal in <H,H^g> implies H normal in <H,g>.
##

InParentFOA("OneElementShowingNotWeaklyNormal", IsGroup,IsGroup, DeclareAttribute);

#############################################################################
##
#O OneElementShowingNotWeaklySPermutable( <G>, <H> ) . . . . for groups
##
## This operation finds an element g of G such that H Sper <H, H^g>
## but H not Sper <H,g> if such an element exists, fail otherwise.
## H is S-permutable in G when H Sper <H,H^g> implies H Sper <H,g>.
##
InParentFOA("OneElementShowingNotWeaklySPermutable", IsGroup,IsGroup, DeclareAttribute);

#############################################################################
##
#O IsWeaklyNormal( <G>, <H> ) . . . . for groups
##
## This operation returns true if a subgroup H of G is
## weakly normal, false otherwise.
## H is weakly normal when H^g\le N_G(H) implies g\in N_G(H)
##
InParentFOA("IsWeaklyNormal", IsGroup,IsGroup, DeclareProperty);

#############################################################################
##
#O IsWeaklySPermutable( <G>, <H> ) . . . . for groups
##
## This operation returns true if a subgroup H of G is
## weakly S-permutable, false otherwise.
## H is weakly S-permutable in G when H Sper <H,H^g> implies H Sper <H,g>.
##
InParentFOA("IsWeaklySPermutable", IsGroup,IsGroup, DeclareProperty);

#############################################################################
##
#O OneElementShowingNotWeaklyPermutable( <G>, <H> ) . . . . for groups
##
## This operation finds an element g such that H per <H, H^g> but
## H not per <H, g> if such an element exists, fail otherwise.
## H is permutable in G when H per <H,H^g> implies H per <H,g>.
##
InParentFOA("OneElementShowingNotWeaklyPermutable", IsGroup,IsGroup, DeclareAttribute);
#############################################################################
##
#O IsWeaklyPermutable( <G>, <H> ) . . . . for groups
##
## This operation returns true if a subgroup H of G is
## weakly permutable, false otherwise.
## H is weakly permutable in G when H per <H, H^g> implies H per <H, g>.
##
InParentFOA("IsWeaklyPermutable", IsGroup,IsGroup, DeclareProperty);

#############################################################################
##
#A OneSubnormalNonPermutableSubgroup( <G> ) . . . . for finite groups
##
## The default method finds a non conjugate-permutable subnormal subgroup
## of G if there exists one such subgroup, it returns fail otherwise.
##

DeclareAttribute("OneSubnormalNonPermutableSubgroup", IsGroup);

#############################################################################
##
#O OneSubgroupInWhichSubnormalNotNormal( <G>, <H> ) . . . . for groups
##
## This operation finds a subgroup K of G such that H is subnormal in K,
## but H is not normal in K if such a subgroup exists, it returns fail otherwise.
## H satisfies the subnormalizer condition in G when H subnormal in K
## implies that H is normal in K.
##

InParentFOA("OneSubgroupInWhichSubnormalNotNormal", IsGroup,IsGroup, DeclareAttribute);

#############################################################################
##
#O IsWithSubnormalizerCondition( <G>, <H> ) . . . . for groups
##
## This operation returns true if H satisfies the subnormaliser
## condition in G, false otherwise.
## H satisfies the subnormalizer condition in G when H subnormal in K
## implies that H is normal in K.
##

InParentFOA("IsWithSubnormalizerCondition", IsGroup,IsGroup, DeclareProperty);
Transatlantic(IsWithSubnormalizerCondition);
Transatlantic(IsWithSubnormalizerConditionOp);
Transatlantic(IsWithSubnormalizerConditionInParent);

#############################################################################
##
#O OneSubgroupInWhichSubnormalNotPermutable( <G>, <H> ) . . . . for groups
##
## This operation finds a subgroup K of G such that H is subnormal in K,
## but H is not permutable in K if such a subgroup exists,
## otherwise it returns fail.
## H satisfies the subpermutizer condition in G when H subnormal in K
## implies that H is permutable in K.
##

InParentFOA("OneSubgroupInWhichSubnormalNotPermutable", IsGroup,IsGroup, DeclareAttribute);

#############################################################################
##
#O IsWithSubpermutizerCondition( <G>, <H> ) . . . . for groups
##
## This operation returns true if H satisfies the subpermutiser condition
## in G, otherwise it returns false.
## H satisfies the subpermutizer condition in G when H subnormal in K
## implies that H is permutable in K.
##

InParentFOA("IsWithSubpermutizerCondition", IsGroup,IsGroup, DeclareProperty);
DeclareSynonym("IsWithSubpermutiserCondition", IsWithSubpermutizerCondition);
DeclareSynonym("IsWithSubpermutiserConditionOp", IsWithSubpermutizerConditionOp);
DeclareSynonymAttr("IsWithSubpermutiserConditionInParent", IsWithSubpermutizerConditionInParent);

#############################################################################
##
#O OneSubgroupInWhichSubnormalNotSPermutable( <G>, <H> ) . . . . for groups
##
## This operation finds a subgroup K of G such that H is subnormal in K,
## but H is not S-permutable in K if such a subgroup exists,
## otherwise it returns fail.
## H satisfies the S-subpermutizer condition in G when H subnormal in K
## implies that H is S-permutable in K.
##

InParentFOA("OneSubgroupInWhichSubnormalNotSPermutable", IsGroup,IsGroup, DeclareAttribute);

#############################################################################
##
#O IsWithSSubpermutizerCondition( <G>, <H> ) . . . . for groups
##
## This operation returns true if H satisfies the S-subpermutiser
## condition in G, otherwise it returns false.
## H satisfies the S-subpermutiser condition in G when H subnormal in K
## implies that H is S-permutable in K.
##

InParentFOA("IsWithSSubpermutizerCondition", IsGroup,IsGroup, DeclareProperty);

DeclareSynonym("IsWithSSubpermutiserCondition", IsWithSSubpermutizerCondition);
DeclareSynonym("IsWithSSubpermutiserConditionOp", IsWithSSubpermutizerConditionOp);
DeclareSynonymAttr("IsWithSSubpermutiserConditionInParent", IsWithSSubpermutizerConditionInParent);

#############################################################################
##
#F IsAbCp( <G>, ) . . . . for finite groups
##
## Returns true if the group has an abelian Sylow p-subgroup P and every
## subgroup of P is normalised by N_G(P), false otherwise.
##

KeyDependentOperation("IsAbCp", IsGroup, IsPosInt, "prime");

#############################################################################
##
#F IsCp( <G>, ) . . . . for finite groups
##
## Returns true if every subgroup of a Sylow p-subgroup P of G
## is normalised by N_G(P), false otherwise.
##

KeyDependentOperation("IsCp", IsGroup, IsPosInt, "prime");

#############################################################################
##
#F IsXp( <G>, ) . . . . for finite groups
##
## Returns true if every subgroup of a Sylow p-subgroup P of G
## is permutable in N_G(P), false otherwise.
##

KeyDependentOperation("IsXp", IsGroup, IsPosInt, "prime");

#############################################################################
##
#F IsYp( <G>, ) . . . . for finite groups
##
## Returns true if whenever $H\le K$ are two p-subgroups,
## H is S-permutable in N_G(K), false otherwise.
##

KeyDependentOperation("IsYp", IsGroup, IsPosInt, "prime");

#############################################################################
##
#F IsDedekindSylow( <G>, ) . . . . for finite groups
##
## Returns true if a Sylow p-subgroup of G is Dedekind, else returns false
##

DeclareGlobalFunction( "IsDedekindSylow" );

#############################################################################
##
#F IsIwasawaSylow( <G>, ) . . . . for finite groups
##
## Returns true if a Sylow p-subgroup of G is Iwasawa, else returns false
##

DeclareGlobalFunction( "IsIwasawaSylow" );

#############################################################################
##
#P IsPTGroup( <G> ) . . . . for finite groups
##
## Returns true if the group is a PT-group, else returns false
##

DeclareProperty("IsPTGroup",IsGroup);

#############################################################################
##
#A OneSubnormalNonSPermutableSubgroup( <G> ) . . . . for finite groups
##
## The default method finds a non-S-permutable subnormal subgroup of G
## if such a subgroup exists, fail otherwise.
##

DeclareAttribute("OneSubnormalNonSPermutableSubgroup", IsGroup);

#############################################################################
##
#A OneSubnormalNonSNPermutableSubgroup( <G> ) . . . . for finite groups
##
## The default method finds a non SN-permutable subnormal
## (permutable with all system normalisers) subgroup
## of the soluble group G if such a subgroup exists, fail otherwise.
##

DeclareAttribute("OneSubnormalNonSNPermutableSubgroup", IsGroup);

#############################################################################
##
#P IsPSTGroup( <G> ) . . . . for finite groups
##
## Returns true if the group is a PST-group, else returns false
##

DeclareProperty("IsPSTGroup",IsGroup);

#############################################################################
##
#P IsPSNTGroup( <G> ) . . . . for finite groups
##
## Returns true if the soluble group is a PSNT-group, else returns false
## A soluble group is a PSNT-group if permutability with system
## normalisers is transitive.
##

DeclareProperty("IsPSNTGroup",IsGroup);
#############################################################################
##
#A OneSubnormalNonConjugatePermutableSubgroup( <G> ) . . . . for finite groups
##
## The default method finds a non conjugate-permutable subnormal subgroup
## of G if such a subgroup exists, fail otherwise.
##

DeclareAttribute("OneSubnormalNonConjugatePermutableSubgroup", IsGroup);

#############################################################################
##
#P IsCPTGroup( <G> ) . . . . for finite groups
##
## Returns true if the group is a CPT-group (every subnormal subgroup
## is conjugate-permutable), else returns false
##

DeclareProperty("IsCPTGroup",IsGroup);

#############################################################################
##
#A OneSubnormalNonNormalSubgroup( <G> ) . . . . for finite groups
##
## The default method finds a non-normal subnormal subgroup of G
## if such a subgroup exists, otherwise returns false.
##

DeclareAttribute("OneSubnormalNonNormalSubgroup", IsGroup);

#############################################################################
##
#P IsTGroup( <G> ) . . . . for finite groups
##
## Returns true if the group is a T-group, else returns false
##

DeclareProperty("IsTGroup",IsGroup);

#############################################################################
##
#A IwasawaTriple( <G> ) . . . . for finite groups
##
## Computes an Iwasawa triple for the p-group G
##

DeclareAttribute("IwasawaTriple", IsGroup);

#############################################################################
##
#P IsSCGroup( <G> ) . . . . for finite groups
##
## Returns true if the group is an SC-group, and false otherwise.
## A group is an SC-group if all its chief factors are simple.
##
##
DeclareProperty("IsSCGroup",IsGroup);

#############################################################################
##
#F IwasawaTripleWithSubgroup( <g>, <x>, ) . . . . for finite groups
##
## Returns an Iwasawa triple of the p-group g such that x belongs to it
## if it exists one, fail otherwise.
##

DeclareGlobalFunction( "IwasawaTripleWithSubgroup" );

#############################################################################
##
#F AllGeneratorsCyclicPGroup( <x>, p ) . . . . for a p-element <x> of a group
##
## This operation computes all generators of the cyclic group
## generated by the p-element <x>.
##
##

DeclareGlobalFunction( "AllGeneratorsCyclicPGroup" );

#############################################################################
##
#O AreMutuallyPermutableSubgroups( <G>, , <V> ) . . . . for finite groups
#O AreMutuallyPermutableSubgroups( , <V> )
##
## This operation returns true if and <V> are mutually permutable,
## otherwise it returns false.
## Two subgroups and <V> are mutually permutable
## if permutes with all subgroups of <V>
## and <V> permutes with all subgroups of .
##

AreMutuallyPermutableSubgroups:=NewOperation("AreMutuallyPermutableSubgroups",
 [IsGroup,IsGroup,IsGroup]);

#############################################################################
##
#O AreTotallyPermutableSubgroups( <G>, , <V> ) . . . . for finite groups
#O AreTotallyPermutableSubgroups( , <V> )
##
## This operation returns true if and <V> are totally permutable,
## otherwise it returns false.
## Two subgroups and <V> are totally permutable
## if every subgroup of permutes with every subgroup of <V>.
##

AreTotallyPermutableSubgroups:=NewOperation("AreTotallyPermutableSubgroups",
 [IsGroup,IsGroup,IsGroup]);

#############################################################################
##
#O OnePairShowingNotMutuallyPermutableSubgroups( <G>, , <V> ) . . . . for finite groups
#O OnePairShowingNotMutuallyPermutableSubgroups( , <V> )
##
## This function returns a pair [U, C] or a pair [D, V] such that
## U does not permute with C or D does not permute with V if such a pair
## exists, fail otherwise.
##

OnePairShowingNotMutuallyPermutableSubgroups:=NewOperation("OnePairShowingNotMutuallyPermutableSubgroups",
 [IsGroup,IsGroup,IsGroup]);

#############################################################################
##
#O OnePairShowingNotTotallyPermutableSubgroups( <G>, , <V> ) . . . . for finite groups
#O OnePairShowingNotTotallyPermutableSubgroups( , <V> )
##
## Returns a pair [C,D] such that C and D do not permute,
## with C subgroup of U and D subgroup of V if such a pair exists,
## fail otherwise.
##

OnePairShowingNotTotallyPermutableSubgroups:=NewOperation("OnePairShowingNotTotallyPermutableSubgroups",
 [IsGroup,IsGroup,IsGroup]);

#############################################################################
##
#O AreMutuallyFPermutableSubgroups( <G>, , <V>, <FU>, <FV> ) . . . . for finite groups
#O AreMutuallyFPermutableSubgroups( , <V>, <FU>, <FV> )
##
## This operation returns true if permutes with <V> and all members
## of <FV> and <V> permutes with and all members of <FU>. Here <FU>
## and <FV> are lists of subgroups of and <V>, respectively.
##

AreMutuallyFPermutableSubgroups:=NewOperation("AreMutuallyFPermutableSubgroups",
 [IsGroup,IsGroup,IsGroup,IsList, IsList]);

#############################################################################
##
#O AreTotallyFPermutableSubgroups( <G>, , <V>, <FU>, <FV> ) . . . . for finite groups
#O AreTotallyFPermutableSubgroups( , <V>, <FU>, <FV> )
##
## This operation returns true if and all members of <FU> permute
## with <V> and all members of <FV>. Here <FU> and <FV> are lists of
## subgroups of and <V>, respectively.
##

AreTotallyFPermutableSubgroups:=NewOperation("AreTotallyFPermutableSubgroups",
 [IsGroup,IsGroup,IsGroup,IsList,IsList]);

#############################################################################
##
#O OnePairShowingNotMutuallyFPermutableSubgroups( <G>, , <V>, <FU>, <FV> ) . . . . for finite groups
#O OnePairShowingNotMutuallyFPermutableSubgroups( , <V>, <FU>, <FV> )
##
## This function returns a pair [U, C] or a pair [D, V] such that
## U does not permute with V or a member of FV or V does not permute
## with U or a member of FU if such a pair exixts; otherwise,
## it returns fail. Here FU and FV are lists of subgroups of
## U and V, respectively.
##

OnePairShowingNotMutuallyFPermutableSubgroups:=NewOperation("OnePairShowingNotMutuallyFPermutableSubgroups",
 [IsGroup,IsGroup,IsGroup,IsList,IsList]);

#############################################################################
##
#O OnePairShowingNotTotallyFPermutableSubgroups( <G>, , <V>, <FU>, <FV> ) . . . . for finite groups
#O OnePairShowingNotTotallyFPermutableSubgroups( , <V>, <FU>, <FV> )
##
## Returns a pair [C,D] with C in FU or C=U and D in FV or D=V such that
## C and D do not permute, where FU and FV are lists of subgroups of
## U and V, respectively, if such a pair exists; otherwise, it returns fail.
##

OnePairShowingNotTotallyFPermutableSubgroups:=NewOperation("OnePairShowingNotTotallyFPermutableSubgroups",
 [IsGroup,IsGroup,IsGroup,IsList,IsList]);

#############################################################################
##
## True methods
##
##

InstallTrueMethod(IsCPTGroup,IsPTGroup);
InstallTrueMethod(IsConjugatePermutableInParent,IsPermutableInParent);
InstallTrueMethod(IsPSNTGroup,IsNilpotentGroup);
InstallTrueMethod(IsPSNTGroup,IsPTGroup);
InstallTrueMethod(IsPSTGroup,IsNilpotentGroup);
InstallTrueMethod(IsPSTGroup,IsPTGroup);
InstallTrueMethod(IsPSTGroup,IsTGroup);
InstallTrueMethod(IsPTGroup,IsTGroup);
InstallTrueMethod(IsPermutableInParent,IsNormalInParent);
InstallTrueMethod(IsSCGroup, IsPSTGroup);
InstallTrueMethod(IsSNPermutableInParent,IsPermutableInParent);
InstallTrueMethod(IsSPermutableInParent,IsPermutableInParent);
InstallTrueMethod(IsSupersolvableGroup, IsPSNTGroup and IsSolvableGroup);
InstallTrueMethod(IsSupersolvableGroup, IsSolvableGroup and IsSCGroup);
InstallTrueMethod(IsTGroup,IsGroup and IsAbelian);
InstallTrueMethod(IsWeaklyNormalInParent,IsNormalInParent);
InstallTrueMethod(IsWeaklyPermutableInParent,IsPermutableInParent);
InstallTrueMethod(IsWeaklyPermutableInParent,IsWeaklyNormalInParent);
InstallTrueMethod(IsWeaklySPermutableInParent, IsWeaklyPermutableInParent);
InstallTrueMethod(IsWeaklySPermutableInParent,IsSPermutableInParent);
InstallTrueMethod(IsWithSSubpermutizerConditionInParent,IsWeaklySPermutableInParent);
InstallTrueMethod(IsWithSSubpermutizerConditionInParent,IsWithSubpermutizerConditionInParent);
InstallTrueMethod(IsWithSubnormalizerConditionInParent,IsWeaklyNormalInParent);
InstallTrueMethod(IsWithSubpermutizerConditionInParent,IsWeaklyPermutableInParent);
InstallTrueMethod(IsWithSubpermutizerConditionInParent,IsWithSubnormalizerConditionInParent);

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