gap Notes

how to find all subgroups of a group

ischarsimple:=function(G)
    local normal,flag,i;
    flag:=true;
    normal:=NormalSubgroups(G);
        for i in [2 .. Length(normal)-1] do
            if(IsCharacteristicSubgroup(G,normal[i])) then
            flag:=false;
            break;
            fi;
            od;
        return flag;


    end;

%gap
num:=List([1 .. 65],NumberSmallGroups);
    for order in [1 .. 65] do
    for i in  [1 .. num[order]] do
     if ischarsimple(SmallGroup(order,i)) then
        Print(order," , ");
        break;
     fi;
        od;
        od;

经计算，最小非交换特征单群阶数为60

有单群的阶数:2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 43 , 47 , 53 ,
59 , 60 , 61 ,

有特征单群的阶数:1 , 2 , 3 , 4 , 5 , 7 , 8 , 9 , 11 , 13 , 16 , 17 , 19 , 23 , 25 , 27 , 29 ,
31 , 32 , 37 , 41 , 43 , 47 , 49 , 53 , 59 , 60 , 61 , 64 ,

问题:是特征单群单不是单群的非交换群的最小阶数是多少？

IsomorphismPermGroup

returns an isomorphism from the group G onto a permutation group which is isomorphic to G. The method will select a suitable permutation representation.

iso:=IsomorphismPermGroup(g)返回g到置换群的同构
Image(iso,g);返回g与同构的置换群

SmallerDegreePermutationRepresentation 将置换群表示成最小阶集合上的忠实作用

StructureDescription

FactorGroup 商群

NormalSubgroups

NaturalHomomorphismByNormalSubgroup 自然同态

IsomorphismGroups 计算同构，如果不同构返回fail

Image(IsomorphismSimplifiedFpGroup(Image(IsomorphismFpGroupByPcgs( FamilyPcgs(CyclicGroup(8)),"g"))));  Pc群转为自由群

RelatorsOfFpGroup(g)

SetReducedMultiplication 化简自由群商群的元素乘法

 G:=F/ParseRelators(GeneratorsOfGroup(F),"a^4=b^2=1,b^-1*a*b=a^-1");
 SetReducedMultiplication(G);

a=G.1;b=G.2
gap> a*b^3;
a*b

IsomorphicSubgroups(G,H)

 AsGroup

singtren@LAPTOP-4N5MIIBA:/mnt/c/Users/sinren$ gap
 ┌───────┐   GAP 4.10.2 of 19-Jun-2019
 │  GAP  │   https://www.gap-system.org
 └───────┘   Architecture: x86_64-pc-linux-gnu-default64-kv3
 Configuration:  gmp 6.1.2
 Loading the library and packages ...
 Packages:   AClib 1.3.1, Alnuth 3.1.1, AtlasRep 2.1.0, AutoDoc 2019.05.20, AutPGrp 1.10, Carat 2.2.3, CRISP 1.4.4,
             Cryst 4.1.19, CrystCat 1.1.9, CTblLib 1.2.2, FactInt 1.6.2, FGA 1.4.0, Forms 1.2.5, GAPDoc 1.6.2,
             genss 1.6.5, IO 4.6.0, IRREDSOL 1.4, LAGUNA 3.9.3, orb 4.8.2, Polenta 1.3.8, Polycyclic 2.14,
             PrimGrp 3.3.2, RadiRoot 2.8, recog 1.3.2, ResClasses 4.7.2, SmallGrp 1.3, Sophus 1.24, SpinSym 1.5.1,
             TomLib 1.2.8, TransGrp 2.0.4, utils 0.63
 Try '??help' for help. See also '?copyright', '?cite' and '?authors'
gap> F:=FreeGroup("a","b");
<free group on the generators [ a, b ]>
gap> G:=F/Parse
    ParseArguments
    ParseBackwards
    ParseBackwardsWithPrefix
    ParseBibFiles
    ParseBibStrings
    ParseBibXMLextFiles
    ParseBibXMLextString
    ParseError
    ParseForwards
    ParseForwardsWithSuffix
    ParseRelators
    ParseTestFile
    ParseTestInput
    ParseTreeXMLFile
    ParseTreeXMLString
gap> G:=F/ ParseRelators(GeneratorsOfGroup(F),"h^4=k^4=1,h^2=k^2,k^-1*h*k=h^-1");
Error, missing generator h at /mnt/d/Download/gap-4.10.2/lib/wordass.gi:1084 called from
DoValWord( 1 ) at /mnt/d/Download/gap-4.10.2/lib/wordass.gi:1152 called from
PPValWord( gens, nams, b[i] ) at /mnt/d/Download/gap-4.10.2/lib/wordass.gi:1220 called from
<function "ParseRelators">( <arguments> )
 called from read-eval loop at *stdin*:2
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk> quitl
> quit;
Syntax warning: Unbound global variable in *errin*:2
quit;
^^^^
Error, Variable: 'quitl' must have a value in
  <corrupted statement>  called from
DoValWord( 1 ) at /mnt/d/Download/gap-4.10.2/lib/wordass.gi:1152 called from
PPValWord( gens, nams, b[i] ) at /mnt/d/Download/gap-4.10.2/lib/wordass.gi:1220 called from
<function "ParseRelators">( <arguments> )
 called from read-eval loop at *errin*:2
Syntax error: ; expected in *errin*:2
quit;
^^^^
brk> quit;
gap> F:=FreeGroup("h","k");
<free group on the generators [ h, k ]>
gap> G:=F/ ParseRelators(GeneratorsOfGroup(F),"h^4=k^4=1,h^2=k^2,k^-1*h*k=h^-1");
<fp group on the generators [ h, k ]>
gap> Size(G);
8
gap> StructureDescription(G);
"Q8"
gap> F:=FreeGroup("a","b");
<free group on the generators [ a, b ]>
gap> G:=F/ParseRelators(GeneratorsOfGroup(F),"a^4=b^2=1,b^-2*a*b=a^-1};
Syntax error: String must not include <newline>
G:=F/ParseRelators(GeneratorsOfGroup(F),"a^4=b^2=1,b^-2*a*b=a^-1};
                                        ^^^^^^^^^^^^^^^^^^^^^^^^^^
> G:=F/ParseRelators(GeneratorsOfGroup(F),"a^4=b^2=1,b^-1*a*b=a^-1");
Syntax error: ) expected
G:=F/ParseRelators(GeneratorsOfGroup(F),"a^4=b^2=1,b^-1*a*b=a^-1");
^
gap> G:=F/ParseRelators(GeneratorsOfGroup(F),"a^4=b^2=1,b^-1*a*b=a^-1");
<fp group on the generators [ a, b ]>
gap> Size(G);
8
gap> Centr
    CentralAutos
    CentralAutosNL
    CentralCharacter
    CentralIdempotentsOfAlgebra
    CentralIdempotentsOfSemiring
    CentralNormalSeriesByPcgs
    CentralRelations
    CentralStepClEANS
    CentralStepConjugatingElement
    CentralStepRatClPGroup
    Centraliser
    CentraliserInFiniteDimensionalAlgebra
    CentraliserInGLnZ
    CentraliserInParent
    CentraliserModulo
    CentraliserNormalCSPG
    CentraliserNormalTransCSPG
    CentraliserOp
    CentraliserSizeLimitConsiderFunction
    CentraliserTransSymmCSPG
    CentraliserWreath
    CentralizeByCentralSeries
    Centralizer
    CentralizerAffineCrystGroup
    CentralizerByCentralLayer
    CentralizerBySeries
    CentralizerElement
    CentralizerInAssociativeGaussianMatrixAlgebra
    CentralizerInFiniteDimensionalAlgebra
    CentralizerInGLnZ
    CentralizerInParent
    CentralizerModulo
    CentralizerNilpotentPcpGroup
    CentralizerNormalCSPG
    CentralizerNormalTransCSPG
    CentralizerOp
    CentralizerOrder
    CentralizerPcpGroup
    CentralizerPointGroupInGLnZ
    CentralizerSizeLimitConsiderFunction
    CentralizerSolvableGroup
    CentralizerTransSymmCSPG
    CentralizerWreath
    CentralizesLayer
    Centre
    CentreFromSCTable
    CentreNilpotentPcpGroup
    CentreOfCharacter
    CentrePcGroup
    CentrePcpGroup
gap> ?Centre
Help: several entries match this topic - type ?2 to get match [2]

[1] Reference: center
[2] Reference: Center                                                                                                  [3] Reference: Centre                                                                                                  [4] HAP (not loaded): Centre
[5] HAP (not loaded): Centre
[6] loops (not loaded): center
[7] loops (not loaded): Center                                                                                         [8] polycyclic: Centre                                                                                                 [9] Reference: Centre for groups with pcgs                                                                             [10] Reference: centre of a character
[11] Reference: CentreOfCharacter                                                                                      [12] QPA (not loaded): Centre/Center                                                                                   [13] XGAP (not loaded): Centres
[14] XMod (not loaded): CentreXMod                                                                                     gap> ?2
  35.4-5 Centre

  ‣ Centre( M ) ─────────────────────────────────────────────────────────────────────────────────────────── attribute    ‣ Center( M ) ─────────────────────────────────────────────────────────────────────────────────────────── attribute  
  Centre  returns the centre of the magma M, i.e., the domain of those elements m ∈ M that commute and associate with   all elements of M. That is, the set { m ∈ M; ∀ a, b ∈ M: ma = am, (ma)b = m(ab), (am)b = a(mb), (ab)m = a(bm) }.  
  Center is just a synonym for Centre.                                                                                 
  For associative magmas we have that Centre( M ) = Centralizer( M, M ), see Centralizer (35.4-4).                     
  The  centre  of  a  magma  is  always commutative (see IsCommutative (35.4-9)). (When one installs a new method for    Centre,  one  should  set the IsCommutative (35.4-9) value of the result to true, in order to make this information    available.)

  35.4-6 Idempotents                                                                                                   
  ‣ Idempotents( M ) ────────────────────────────────────────────────────────────────────────────────────── attribute  
  The set of elements of M which are their own squares.                                                                
  35.4-7 IsAssociative                                                                                                 
  ‣ IsAssociative( M ) ───────────────────────────────────────────────────────────────────────────────────── property  
  A magma M is associative if for all elements a, b, c ∈ M the equality (a * b) * c = a * (b * c) holds.              gap> Centre
    Centre
    CentreFromSCTable
    CentreNilpotentPcpGroup
    CentreOfCharacter
    CentrePcGroup
    CentrePcpGroup
gap> Centre(G);
Group([ a^2 ])
gap> AsList(last);
[ <identity ...>, <[ [ 1, 1 ] ]|a^-2> ]
gap> Group([ a^2 ]);
Error, Variable: 'a' must have a value
not in any function at *stdin*:15
gap> G.1
> ;
a
gap> G.2;
b
gap> G.1*G.2;
a*b
gap> G.1^3*G.2;
a^3*b
gap> G.1^4*G.2;
a^4*b
gap> a in G;
Error, Variable: 'a' must have a value
not in any function at *stdin*:22
gap> a=G.1;b=G.2;
Error, Variable: 'a' must have a value
not in any function at *stdin*:23
Error, Variable: 'b' must have a value
not in any function at *stdin*:23
gap> a:=G.1;b:=G.2;
a
b
gap> a^2;
a^2
gap> a^4;
a^4
gap> a^4=a^0;
true
gap> a^2*b=b^2*a;
false
gap> a^2*b=b*a^2;
true
gap> a^2*b*a=b*a*a^2;
true
gap> a^2*b*a^2=b*a^2*a^2;
true
gap> a^2*b*a^3=b*a^3*a^2;
true
gap> SetReducedMultiplication(b*a^3*a^2);
gap> b*a^3*a^2;
b*a
gap> a^4;
<identity ...>
gap> a*b^3;
a*b
gap> a^2;
a^2
gap> a^2*b
> ;
b*a^2
gap> SetReducedMultiplication(G);
gap> a*b*a*a*b*b^2;
a^-1
gap> a^b;
a^-1
gap> b^a;
b*a^2
gap> Centre(G);
Group([ a^2 ])
gap> UnderlyingElement(a);
a
gap> ElementOfFpGroup(G,a);
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method found for `ElementOfFpGroup' on 2 arguments at /mnt/d/Download/gap-4.10.2/lib/methsel2.g:250 called from
<function "HANDLE_METHOD_NOT_FOUND">( <arguments> )
 called from read-eval loop at *stdin*:46
type 'quit;' to quit to outer loop
brk> quit;
gap> G
> G;
Syntax error: ; expected
G;
^
gap> G;
<fp group of size 8 on the generators [ a, b ]>
gap> AsList(G);
[ <identity ...>, a, a^-1, b, a^2, a*b, b*a, b*a^2 ]
gap> Subgroup(G,[a^2]);
Group([ a^2 ])
gap> AsList(last)l
Syntax error: ; expected
AsList(last)l
            ^
gap> AsList(last);
[ <identity ...>, <[ [ 1, 1 ] ]|a^-2> ]
gap> Size(Centre(G));
2
gap> a^a;
a
gap> a^b;
a^-1
gap> F1:=FreeGroup("a","b","x");
<free group on the generators [ a, b, x ]>
gap> r:=ParseRelators([a,b,x],"a^x=a,b^x=b,a^4=b^2=1,a^b=a^-1,x^4=1,a^2=x^2");
Error, Variable: 'x' must have a value
not in any function at *stdin*:57
gap> Unbind(a,b);
Syntax error: 'Unbind': argument should be followed by ')'
Unbind(a,b);
        ^
gap> r:=ParseRelators(GeneratorsOf,"a^x=a,b^x=b,a^4=b^2=1,a^b=a^-1,x^4=1,a^2=x^2");
    GeneratorsOfAdditiveGroup
    GeneratorsOfAdditiveMagma
    GeneratorsOfAdditiveMagmaWithInverses
    GeneratorsOfAdditiveMagmaWithZero
    GeneratorsOfAlgebra
    GeneratorsOfAlgebraModule
    GeneratorsOfAlgebraWithOne
    GeneratorsOfCentralizerOfPcp
    GeneratorsOfDivisionRing
    GeneratorsOfDomain
    GeneratorsOfEquivalenceRelationPartition
    GeneratorsOfExtASet
    GeneratorsOfExtLSet
    GeneratorsOfExtRSet
    GeneratorsOfExtUSet
    GeneratorsOfFLMLOR
    GeneratorsOfFLMLORWithOne
    GeneratorsOfField
    GeneratorsOfGroup
    GeneratorsOfIdeal
    GeneratorsOfInverseMonoid
    GeneratorsOfInverseSemigroup
    GeneratorsOfLayer
    GeneratorsOfLeftIdeal
    GeneratorsOfLeftMagmaIdeal
    GeneratorsOfLeftModule
    GeneratorsOfLeftOperatorAdditiveGroup
    GeneratorsOfLeftOperatorRing
    GeneratorsOfLeftOperatorRingWithOne
    GeneratorsOfLeftVectorSpace
    GeneratorsOfMagma
    GeneratorsOfMagmaIdeal
    GeneratorsOfMagmaWithInverses
    GeneratorsOfMagmaWithOne
    GeneratorsOfMonoid
    GeneratorsOfNearAdditiveGroup
    GeneratorsOfNearAdditiveMagma
    GeneratorsOfNearAdditiveMagmaWithInverses
    GeneratorsOfNearAdditiveMagmaWithZero
    GeneratorsOfPcp
    GeneratorsOfPresentation
    GeneratorsOfReesMatrixSemigroup
    GeneratorsOfReesMatrixSemigroupNC
    GeneratorsOfReesZeroMatrixSemigroup
    GeneratorsOfReesZeroMatrixSemigroupNC
    GeneratorsOfRightIdeal
    GeneratorsOfRightMagmaIdeal
    GeneratorsOfRightModule
    GeneratorsOfRightOperatorAdditiveGroup
    GeneratorsOfRing
    GeneratorsOfRingForIdeal
    GeneratorsOfRingWithOne
    GeneratorsOfRws
    GeneratorsOfSemigroup
    GeneratorsOfSemiring
    GeneratorsOfSemiringWithOne
    GeneratorsOfSemiringWithOneAndZero
    GeneratorsOfSemiringWithZero
    GeneratorsOfTwoSidedIdeal
    GeneratorsOfVectorSpace
gap> r:=ParseRelators(GeneratorsOfGroup(F1),"a^x=a,b^x=b,a^4=b^2=1,a^b=a^-1,x^4=1,a^2=x^2");
[ x^-1*a*x*a^-1, x^-1*b*x*b^-1, a^4, b^2, a^-1*b^-1*a^-1*b, x^4, x^2*a^-2 ]
gap> G1:=F1/r;
<fp group on the generators [ a, b, x ]>
gap> Size(G);
8
gap> a^-1*b*a;
b*a^2
gap> a*b*a^-1;
b*a^2
gap> Structur
    StructuralCopy
    StructureConstantsPadicNumbers
    StructureConstantsTable
    StructureDescription
    StructureDescriptionCharacterTableName
    StructureDescriptionForAbelianGroups
    StructureDescriptionForFiniteGroups
    StructureDescriptionForFiniteSimpleGroups
    StructurePartInTree
gap> StructureDescription(G);
"D8"
gap> Size(G1);
16
gap> StructureDescription(G1);
"(C4 x C2) : C2"
gap> G1;
(C4 x C2) : C2
gap> AsList(G1);
[ <identity ...>, a, a^-1, b, x, x^-1, a^2, a*b, a*x, a*x^-1, a^-1*b, b*x, b*x^-1, a^2*b, a*b*x, a*b*x^-1 ]
gap> IsomorphismSimplifiedFpGroup(G1);
[ a, b, x ] -> [ a, b, x ]
gap> Isomorphi
    IsomorphicMatrixField
    IsomorphicSubgroups
    IsomorphismAbelianGroupViaIndependentGenerators
    IsomorphismAbelianGroups
    IsomorphismFpAlgebra
    IsomorphismFpFLMLOR
    IsomorphismFpGroup
    IsomorphismFpGroupByChiefSeries
    IsomorphismFpGroupByChiefSeriesFactor
    IsomorphismFpGroupByCompositionSeries
    IsomorphismFpGroupByGenerators
    IsomorphismFpGroupByGeneratorsNC
    IsomorphismFpGroupByPcgs
    IsomorphismFpGroupBySubnormalSeries
    IsomorphismFpMonoid
    IsomorphismFpMonoidGeneratorsFirst
    IsomorphismFpMonoidInversesFirst
    IsomorphismFpSemigroup
    IsomorphismGroups
    IsomorphismMatrixAlgebra
    IsomorphismMatrixFLMLOR
    IsomorphismMatrixField
    IsomorphismOfMultGroupByFieldEl
    IsomorphismPartialPermMonoid
    IsomorphismPartialPermSemigroup
    IsomorphismPcGroup
    IsomorphismPcpGroup
    IsomorphismPcpGroupFromFpGroupWithPcPres
    IsomorphismPermGroup
    IsomorphismPermGroupImfGroup
    IsomorphismPermGroupOrFailFpGroup
    IsomorphismReesMatrixSemigroup
    IsomorphismReesZeroMatrixSemigroup
    IsomorphismRefinedPcGroup
    IsomorphismSCAlgebra
    IsomorphismSCFLMLOR
    IsomorphismSCRing
    IsomorphismSimplifiedFpGroup
    IsomorphismSolvableSmallGroups
    IsomorphismSpecialPcGroup
    IsomorphismTransformationMonoid
    IsomorphismTransformationSemigroup
    IsomorphismTypeInfoFiniteSimpleGroup
    IsomorphismUpperUnitriMatGroupPcpGroup
gap> iso:=IsomorphismPermGroup(G1);
[ a, b, x ] -> [ (1,2,6,3)(4,7,5,8), (2,3)(7,8), (1,4,6,5)(2,7,3,8) ]
gap> SmallerDegreePermutationRepresentation(Image(iso));
IdentityMapping( Group([ (1,2,6,3)(4,7,5,8), (2,3)(7,8), (1,4,6,5)(2,7,3,8) ]) )
gap> F3:=FreeGroup("a","b");
<free group on the generators [ a, b ]>
gap> F3/ParseRelators(GeneratorsOfGroup(F3),"";^C
Syntax error: ) expected
F3/ParseRelators(GeneratorsOfGroup(F3),"";
                                         ^
> quit;
gap> F3:=FreeGroup("h","k");
<free group on the generators [ h, k ]>
gap> Q8:= F3/ParseRelators(GeneratorsOfGroup(F3),"h^4=1,k^2=1,h^2=k^2,h^k=h^-1");
<fp group on the generators [ h, k ]>
gap> Size(Q8)l
Syntax error: ; expected
Size(Q8)l
        ^
gap> Size(Q8);
4
gap> Q8:= F3/ParseRelators(GeneratorsOfGroup(F3),"h^4=1,h^2=k^2,h^k=h^-1");
<fp group on the generators [ h, k ]>
gap> Size(Q8);
8
gap> Structru
    identifier has no completions
gap> StructureDescription(Q8);
"Q8"
gap> AsList(Q8);
[ <identity ...>, h, h^-1, k, k^-1, h^2, h*k, h*k^-1 ]
gap> Centre(Q8);
Group([ h^2 ])
gap> F4=FreeGroup("h","k","x");
Error, Variable: 'F4' must have a value
not in any function at *stdin*:84
gap> F4:=FreeGroup("h","k","x");
<free group on the generators [ h, k, x ]>
gap> G4:=F4/ParseRelators(GeneratorsOfGroup(F4),"h^4=1,h^2=k^2,h^k=h^-1,x^4=1,h^2=x^2,h^x=h,k^x=k");
<fp group on the generators [ h, k, x ]>
gap> Size(G4);
16
gap> StructreD
    identifier has no completions
gap> StructureDescription(G4);
"(C4 x C2) : C2"
gap> Isomorphi
    IsomorphicMatrixField
    IsomorphicSubgroups
    IsomorphismAbelianGroupViaIndependentGenerators
    IsomorphismAbelianGroups
    IsomorphismFpAlgebra
    IsomorphismFpFLMLOR
    IsomorphismFpGroup
    IsomorphismFpGroupByChiefSeries
    IsomorphismFpGroupByChiefSeriesFactor
    IsomorphismFpGroupByCompositionSeries
    IsomorphismFpGroupByGenerators
    IsomorphismFpGroupByGeneratorsNC
    IsomorphismFpGroupByPcgs
    IsomorphismFpGroupBySubnormalSeries
    IsomorphismFpMonoid
    IsomorphismFpMonoidGeneratorsFirst
    IsomorphismFpMonoidInversesFirst
    IsomorphismFpSemigroup
    IsomorphismGroups
    IsomorphismMatrixAlgebra
    IsomorphismMatrixFLMLOR
    IsomorphismMatrixField
    IsomorphismOfMultGroupByFieldEl
    IsomorphismPartialPermMonoid
    IsomorphismPartialPermSemigroup
    IsomorphismPcGroup
    IsomorphismPcpGroup
    IsomorphismPcpGroupFromFpGroupWithPcPres
    IsomorphismPermGroup
    IsomorphismPermGroupImfGroup
    IsomorphismPermGroupOrFailFpGroup
    IsomorphismReesMatrixSemigroup
    IsomorphismReesZeroMatrixSemigroup
    IsomorphismRefinedPcGroup
    IsomorphismSCAlgebra
    IsomorphismSCFLMLOR
    IsomorphismSCRing
    IsomorphismSimplifiedFpGroup
    IsomorphismSolvableSmallGroups
    IsomorphismSpecialPcGroup
    IsomorphismTransformationMonoid
    IsomorphismTransformationSemigroup
    IsomorphismTypeInfoFiniteSimpleGroup
    IsomorphismUpperUnitriMatGroupPcpGroup
gap> GroupIso
    identifier has no completions
gap> IsomorphismGroups(G2,G4);
Error, Variable: 'G2' must have a value
not in any function at *stdin*:89
gap> G1
> ;
(C4 x C2) : C2
gap> IsomorphismGroups(G1,G4);
[ a, b, x ] -> [ h*x^2, k*x, x ]
gap>  G5:=F4/ParseRelators(GeneratorsOfGroup(F4),"h^4=1,h^2=k^2,h^k=h^-1,x^4=1,h^2=x^2");
<fp group on the generators [ h, k, x ]>
gap> Size(G5);

#I  Coset table calculation failed -- trying with bigger table limit
^CError, user interrupt in
  g[2 * limit] := 0; at /mnt/d/Download/gap-4.10.2/lib/grpfp.gi:1233 called from
TCENUM.CosetTableFromGensAndRels( fgens, grels, fsgens ) at /mnt/d/Download/gap-4.10.2/lib/grpfp.gi:1063 called from   CosetTableFromGensAndRels( fgens, grels, List( trial, UnderlyingElement )
 ) at /mnt/d/Download/gap-4.10.2/lib/grpfp.gi:3769 called from
Attempt( gens ) at /mnt/d/Download/gap-4.10.2/lib/grpfp.gi:3782 called from
FinIndexCyclicSubgroupGenerator( G, infinity ) at /mnt/d/Download/gap-4.10.2/lib/grpfp.gi:3849 called from
<function "unknown">( <arguments> )
 called from read-eval loop at *stdin*:94
you can 'return;'
brk> return;
#I  Coset table calculation failed -- trying with bigger table limit
^CError, user interrupt in
  nrdef := nrdef + 1; at /mnt/d/Download/gap-4.10.2/lib/grpfp.gi:1248 called from
TCENUM.CosetTableFromGensAndRels( fgens, grels, fsgens ) at /mnt/d/Download/gap-4.10.2/lib/grpfp.gi:1063 called from   CosetTableFromGensAndRels( fgens, grels, trial ) at /mnt/d/Download/gap-4.10.2/lib/grpfp.gi:3757 called from
Attempt( gens ) at /mnt/d/Download/gap-4.10.2/lib/grpfp.gi:3782 called from
FinIndexCyclicSubgroupGenerator( G, infinity ) at /mnt/d/Download/gap-4.10.2/lib/grpfp.gi:3849 called from
<function "unknown">( <arguments> )
 called from read-eval loop at *stdin*:94
you can 'return;'
brk> quit;
#I  Options stack has been reset
gap> G3
> ;
Error, Variable: 'G3' must have a value
not in any function at *stdin*:95
gap> G4;
(C4 x C2) : C2
gap> AsList(G4);
[ <identity ...>, h, h^-1, k, k^-1, x, x^-1, h^2, h*k, h*k^-1, h*x, h*x^-1, k*x, k*x^-1, h*k*x, h*k*x^-1 ]
gap> h=G4.1;
Error, Variable: 'h' must have a value
not in any function at *stdin*:98
gap> G4.1
> ;
h
gap> h:=G4.1;
h
gap> k=G4.2;
Error, Variable: 'k' must have a value
not in any function at *stdin*:102
gap> k:=G4.2;
k
gap> x=G4.3;
Error, Variable: 'x' must have a value
not in any function at *stdin*:104
gap> x:=G4.3;
x
gap> c:=a*x;
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method found for `*' on 2 arguments at /mnt/d/Download/gap-4.10.2/lib/methsel2.g:250 called from  <function "HANDLE_METHOD_NOT_FOUND">( <arguments> )
 called from read-eval loop at *stdin*:106
type 'quit;' to quit to outer loop
brk> c:=h*x;
Syntax warning: Unbound global variable in *errin*:1
c:=h*x;
 ^^
h*x
brk> x
> ;
x
brk> x*x;
x^2
brk> h*h;
h^2
brk> h*x;
h*x
brk> c:=h*x;
h*x
brk> c
> ;
h*x
brk> d:=k*x;
Syntax warning: Unbound global variable in *errin*:10
d:=k*x;
 ^^
k*x
brk> d:=k*x;
k*x
brk> return;
'return' cannot be used in this read-eval-print loop
brk> quit;
gap> k*x;
k*x
gap> d:=k*x;
k*x
gap> d
> ;
k*x
gap> c*d;
h*x*k*x
gap> SetReducedMultiplication(G4);
gap> c*d;
k*h
gap> Subgroup(G4,[c,d]);
Group([ h*x, k*x ])
gap> AsList(Subgroup(G4,[c,d]));
[ <identity ...>, <[ [ 2, 1 ] ]|k*h^-1>, k*h, <[ [ 1, 1, 3, -1 ] ]|h^-1*x^-1>, h^2, x*k, x^-1*k, x^-1*h ]
gap> StructureDescription(, Subgroup(G4,[c,d]);
Syntax error: expression expected
StructureDescription(, Subgroup(G4,[c,d]);
                     ^
gap> StructureDescription(Subgroup(G4,[c,d]);
Syntax error: ) expected
StructureDescription(Subgroup(G4,[c,d]);
                                       ^
gap> StructureDescription(Subgroup(G4,[c,d]));
"D8"
gap> CreateMa
    CreateMainPage
    CreateMakeTest
gap> MappingByFunction(G4,G4,x->x^2);
MappingByFunction( (C4 x C2) : C2, (C4 x C2) : C2, function( x ) ... end )
gap> f:= MappingByFunction(G4,G4,x->x^2);
MappingByFunction( (C4 x C2) : C2, (C4 x C2) : C2, function( x ) ... end )
gap> f
> ;
MappingByFunction( (C4 x C2) : C2, (C4 x C2) : C2, function( x ) ... end )
gap> f(h);
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method found for `CallFuncList' on 2 arguments at /mnt/d/Download/gap-4.10.2/lib/methsel2.g:250 called from
<function "HANDLE_METHOD_NOT_FOUND">( <arguments> )
 called from read-eval loop at *stdin*:122
type 'quit;' to quit to outer loop
brk> IsInjective(f);
false
brk> Range(f);
(C4 x C2) : C2
brk> IsBijective(f);
false
brk> Image(f);
[ <identity ...>, h^2 ]
brk> AsList(G4);
[ <identity ...>, h, h^-1, k, k^-1, x, x^-1, h^2, h*k, h*k^-1, h*x, h*x^-1, k*x, k*x^-1, h*k*x, h*k*x^-1 ]
brk> (h^-1)^2;
h^2
brk> List[AsList(G4),x->x^2];
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method found for `[]' on 3 arguments at /mnt/d/Download/gap-4.10.2/lib/methsel2.g:250 called from ErrorNoReturn( no_method_found ); at /mnt/d/Download/gap-4.10.2/lib/methsel2.g:250 called from
<function "HANDLE_METHOD_NOT_FOUND">( <arguments> )
 called from read-eval loop at *errin*:7
type 'quit;' to quit to outer loop
brk_2> quit;
brk> quit;
gap> List(AsList(G4),x->x^2);
[ <identity ...>, h^2, h^2, h^2, h^2, h^2, h^2, <identity ...>, h^2, h^2, <identity ...>, <identity ...>,
  <identity ...>, <identity ...>, <identity ...>, <identity ...> ]
gap> G4
> ;
(C4 x C2) : C2
gap> DirectProduct(G4,G1);
<fp group of size 256 on the generators [ f1, f2, f3, f4, f5, f6 ]>
gap> d:= DirectProduct(G4,G1);
<fp group of size 256 on the generators [ f1, f2, f3, f4, f5, f6 ]>
gap> Embedin
    identifier has no completions
gap> Embedding(d,1);
[ h, k, x ] -> [ f1, f2, f3 ]
gap> Embedd
    EmbeddedConjugates
    Embedding
    EmbeddingConjugates
    EmbeddingWreathInWreath
gap> Embedding(d,2);
[ a, b, x ] -> [ f4, f5, f6 ]
gap> Image(Embedd
    EmbeddedConjugates
    Embedding
    EmbeddingConjugates
    EmbeddingWreathInWreath
gap> Image(Embedding(d,1));
Group([ f1, f2, f3 ])
gap> Intersect(Image(Embedding(d,1)),Image(Embedding(d,1)));
Error, Variable: 'Intersect' must have a value
not in any function at *stdin*:130
gap> Intersect(Image(Embedding(d,1)),Image(Embedding(d,1));
Error, Variable: 'Intersect' must have a value
not in any function at *stdin*:131
Syntax error: ) expected
Intersect(Image(Embedding(d,1)),Image(Embedding(d,1));
                                                     ^
gap> Intersection(Image(Embedding(d,1)),Image(Embedding(d,1));
Syntax error: ) expected
Intersection(Image(Embedding(d,1)),Image(Embedding(d,1));
                                                        ^
gap> Intersection(Image(Embedding(d,1)),Image(Embedding(d,1)));
Group([ f1, f2, f3 ])
gap> Intersection(Image(Embedding(d,1)),Image(Embedding(d,2)));
Group(<fp, no generators known>)
gap> Size(last);
1
gap> AsList( Intersection(Image(Embedding(d,1)),Image(Embedding(d,2))));
[ <identity ...> ]
gap> G1;
(C4 x C2) : C2
gap> G;
D8
gap> G2;
Error, Variable: 'G2' must have a value
not in any function at *stdin*:139
gap> G3;
Error, Variable: 'G3' must have a value
not in any function at *stdin*:140
gap> G1;
(C4 x C2) : C2
gap> IsomorphicSubgroups(Centre(G),Q8);
[  ]
gap> IsomorphicSubgroups(Q8,Centre(G));
[ [ a^2 ] -> [ h^2 ] ]
gap> epi:=IsomorphicSubgroups(Q8,Centre(G));
[ [ a^2 ] -> [ h^2 ] ]
gap> Image(epi);
Error, usage: Image(<map>), Image(<map>,<elm>), Image(<map>,<coll>) at /mnt/d/Download/gap-4.10.2/lib/mapping.gi:198 called from
<function "Image">( <arguments> )
 called from read-eval loop at *stdin*:145
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk> quit;
gap> epi
> ;
[ [ a^2 ] -> [ h^2 ] ]
gap> epi[1];
[ a^2 ] -> [ h^2 ]
gap> Image(epi[1]);
Group([ h^2 ])
gap> D8=G;
Error, Variable: 'D8' must have a value
not in any function at *stdin*:149
gap> D8:=G;
D8
gap> C4=Cycl
    Cycle
    CycleByPosOp
    CycleFromList
    CycleIndex
    CycleIndexOp
    CycleLength
    CycleLengthOp
    CycleLengths
    CycleLengthsOp
    CycleOp
    CycleStructureClass
    CycleStructurePerm
    CycleStructuresGroup
    CycleTransformationInt
    Cycles
    CyclesOfTransformation
    CyclesOp
    CyclicDecomposition
    CyclicExtensionsTom
    CyclicExtensionsTomOp
    CyclicGroup
    CyclicGroupCons
    CyclicallyReducedWord
    Cyclotomic
    CyclotomicField
    CyclotomicPol
    CyclotomicPolynomial
    Cyclotomics
    CyclotomicsFamily
gap> C4=CyclicGroup(4);
Error, Variable: 'C4' must have a value
not in any function at *stdin*:151
gap> C4:=CyclicGroup(4);
<pc group of size 4 with 2 generators>
gap> D8xC4:=DirectProduct(D8,C4);
<group of size 32 with 4 generators>
gap> List(D8xC4,x->x*x);
[ DirectProductElement( [ <identity ...>, <identity> of ... ] ), DirectProductElement( [ <identity ...>, f2 ] ),
  DirectProductElement( [ <identity ...>, <identity> of ... ] ), DirectProductElement( [ <identity ...>, f2 ] ),         DirectProductElement( [ a^2, <identity> of ... ] ), DirectProductElement( [ a^2, f2 ] ),
  DirectProductElement( [ a^2, <identity> of ... ] ), DirectProductElement( [ a^2, f2 ] ),
  DirectProductElement( [ <identity ...>, <identity> of ... ] ), DirectProductElement( [ <identity ...>, f2 ] ),         DirectProductElement( [ <identity ...>, <identity> of ... ] ), DirectProductElement( [ <identity ...>, f2 ] ),         DirectProductElement( [ <identity ...>, <identity> of ... ] ), DirectProductElement( [ <identity ...>, f2 ] ),         DirectProductElement( [ <identity ...>, <identity> of ... ] ), DirectProductElement( [ <identity ...>, f2 ] ),         DirectProductElement( [ <identity ...>, <identity> of ... ] ), DirectProductElement( [ <identity ...>, f2 ] ),         DirectProductElement( [ <identity ...>, <identity> of ... ] ), DirectProductElement( [ <identity ...>, f2 ] ),         DirectProductElement( [ a^2, <identity> of ... ] ), DirectProductElement( [ a^2, f2 ] ),
  DirectProductElement( [ a^2, <identity> of ... ] ), DirectProductElement( [ a^2, f2 ] ),
  DirectProductElement( [ <identity ...>, <identity> of ... ] ), DirectProductElement( [ <identity ...>, f2 ] ),         DirectProductElement( [ <identity ...>, <identity> of ... ] ), DirectProductElement( [ <identity ...>, f2 ] ),         DirectProductElement( [ <identity ...>, <identity> of ... ] ), DirectProductElement( [ <identity ...>, f2 ] ),         DirectProductElement( [ <identity ...>, <identity> of ... ] ), DirectProductElement( [ <identity ...>, f2 ] ) ]      gap>  List(D8xC4,x->x)
> ;
[ DirectProductElement( [ <identity ...>, <identity> of ... ] ), DirectProductElement( [ <identity ...>, f1 ] ),         DirectProductElement( [ <identity ...>, f2 ] ), DirectProductElement( [ <identity ...>, f1*f2 ] ),
  DirectProductElement( [ a, <identity> of ... ] ), DirectProductElement( [ a, f1 ] ), DirectProductElement( [ a,          f2 ] ), DirectProductElement( [ a, f1*f2 ] ), DirectProductElement( [ b, <identity> of ... ] ),
  DirectProductElement( [ b, f1 ] ), DirectProductElement( [ b, f2 ] ), DirectProductElement( [ b, f1*f2 ] ),
  DirectProductElement( [ a^2, <identity> of ... ] ), DirectProductElement( [ a^2, f1 ] ),
  DirectProductElement( [ a^2, f2 ] ), DirectProductElement( [ a^2, f1*f2 ] ), DirectProductElement( [ a*b,
    <identity> of ... ] ), DirectProductElement( [ a*b, f1 ] ), DirectProductElement( [ a*b, f2 ] ),
  DirectProductElement( [ a*b, f1*f2 ] ), DirectProductElement( [ a^-1, <identity> of ... ] ),
  DirectProductElement( [ a^-1, f1 ] ), DirectProductElement( [ a^-1, f2 ] ), DirectProductElement( [ a^-1,
    f1*f2 ] ), DirectProductElement( [ b*a^2, <identity> of ... ] ), DirectProductElement( [ b*a^2, f1 ] ),
  DirectProductElement( [ b*a^2, f2 ] ), DirectProductElement( [ b*a^2, f1*f2 ] ), DirectProductElement( [ b*a,            <identity> of ... ] ), DirectProductElement( [ b*a, f1 ] ), DirectProductElement( [ b*a, f2 ] ),
  DirectProductElement( [ b*a, f1*f2 ] ) ]
gap> IsGroup(last);
false
gap> Group( List(D8xC4,x->x));
<group with 32 generators>
gap> Group( List(D8xC4,x->x^2));
<group with 32 generators>
gap> Size(last);
4
gap> Size(Group( List(D8xC4,x->x)));
32
gap>  Size(Group( [ DirectProductElement( [ b*a, f1*f2 ] )]));
Error, Variable: 'f1' must have a value
not in any function at *stdin*:162
gap>  AsGroup( List(D8xC4,x->x));
<group of size 32 with 3 generators>
gap> AsGroup( List(D8xC4,x->x^2));
<group of size 4 with 2 generators>
gap>  Set(List(D8xC4,x->x^2));
[ DirectProductElement( [ <identity ...>, <identity> of ... ] ), DirectProductElement( [ <identity ...>, f2 ] ),         DirectProductElement( [ a^2, <identity> of ... ] ), DirectProductElement( [ a^2, f2 ] ) ]
gap> Length( Set(List(D8xC4,x->x^2)));
4
gap> IsomorphicSubgroups(Centre(D8),C4);
[  ]
gap> IsomorphicSubgroups(C4,Centre(D8));
[ [ a^2 ] -> [ f2 ] ]
gap> theta:=IsomorphicSubgroups(C4,Centre(D8))[1];
[ a^2 ] -> [ f2 ]
gap> f
> ;
MappingByFunction( (C4 x C2) : C2, (C4 x C2) : C2, function( x ) ... end )
gap> f:=CompositionMap
    CompositionMapping
    CompositionMapping2
    CompositionMapping2General
    CompositionMaps
gap> f:=CompositionMaping(theta,Embedding(D8xC4,2);
Error, Variable: 'CompositionMaping' must have a value
not in any function at *stdin*:172
Syntax error: ) expected
f:=CompositionMaping(theta,Embedding(D8xC4,2);
                                             ^
gap> f:=CompositionMaping(theta,Embedding(D8xC4,2));
Error, Variable: 'CompositionMaping' must have a value
not in any function at *stdin*:173
gap> ,Embedding(D8xC4,2);
Syntax error: expression expected
,Embedding(D8xC4,2);
^
gap> Embedding(D8xC4,2);
MappingByFunction( <pc group of size 4 with
2 generators>, <group of size 32 with 4 generators>, function( elm ) ... end )
gap> theta
> ;
[ a^2 ] -> [ f2 ]
gap> f:=CompositionMaping(theta,Embedding(D8xC4,2));
Error, Variable: 'CompositionMaping' must have a value
not in any function at *stdin*:178
gap> f:=CompositionMaping();
Error, Variable: 'CompositionMaping' must have a value
not in any function at *stdin*:179
gap> f:=CompositionMaping(theta);
Error, Variable: 'CompositionMaping' must have a value
not in any function at *stdin*:180
gap> f:=CompositionMaping2(Embedding(D8xC4,2),theta);
Error, Variable: 'CompositionMaping2' must have a value
not in any function at *stdin*:181
gap> f:=CompositionMaping2(theta,theta);
Error, Variable: 'CompositionMaping2' must have a value
not in any function at *stdin*:182
gap> ?CompositionMaping
Help: no matching entry found
gap> f:=CompositionMapping(theta,Embedding(D8xC4,2));
Error, the range of <com> and the source of <nxt> must be contained in the same family at /mnt/d/Download/gap-4.10.2/lib/mapping.gi:394 called from
<function "CompositionMapping">( <arguments> )
 called from read-eval loop at *stdin*:184
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk> quit;
gap> Range(theta);
<pc group of size 4 with 2 generators>
gap> C4
> ;
<pc group of size 4 with 2 generators>
gap> Source(Embedding(D8xC4,2));
<pc group of size 4 with 2 generators>
gap> f:=CompositionMapping(Embedding(D8xC4,2),theta);
[ a^2 ] -> [ DirectProductElement( [ <identity ...>, f2 ] ) ]
gap> Source(f);
Group([ a^2 ])
gap> Image(f);
<group with 1 generators>
gap> Size(f);
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method found for `Size' on 1 arguments at /mnt/d/Download/gap-4.10.2/lib/methsel2.g:250 called from
<function "HANDLE_METHOD_NOT_FOUND">( <arguments> )
 called from read-eval loop at *stdin*:191
type 'quit;' to quit to outer loop
brk> quit;
gap> Size(Image(f));
2
gap> FactorGroup(D8xC4,Image(f));
Group([ f1, f2, f3, f4 ])
gap> StructureDescription(last);
"C2 x D8"
gap> IsomorphismGroups(FactorGroup(D8xC4,Image(f)),G4);
fail
gap> G4;
(C4 x C2) : C2
gap> f;
[ a^2 ] -> [ DirectProductElement( [ <identity ...>, f2 ] ) ]
gap> f(a^2);
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method found for `CallFuncList' on 2 arguments at /mnt/d/Download/gap-4.10.2/lib/methsel2.g:250 called from
<function "HANDLE_METHOD_NOT_FOUND">( <arguments> )
 called from read-eval loop at *stdin*:197
type 'quit;' to quit to outer loop
brk> quit;
gap> ImagesElm(f,a^2);
RightCoset(<trivial group>,[ <identity ...>, f2 ])
gap> f;
[ a^2 ] -> [ DirectProductElement( [ <identity ...>, f2 ] ) ]
gap> List(Centre(D8),x->ImagesElm(f,x^2)*x^-1)
> ;
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method found for `*' on 2 arguments at /mnt/d/Download/gap-4.10.2/lib/methsel2.g:250 called from  ImagesElm( f, x ^ 2 ) * x ^ -1 at *stdin*:199 called from
func( elm ) at /mnt/d/Download/gap-4.10.2/lib/coll.gi:715 called from
CallFuncList( ListOp, arg ) at /mnt/d/Download/gap-4.10.2/lib/coll.gi:677 called from
<function "List">( <arguments> )
 called from read-eval loop at *stdin*:200
type 'quit;' to quit to outer loop
brk> quit;
gap> ImagesElm(f,x^2);
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method found for `ImagesElm' on 2 arguments at /mnt/d/Download/gap-4.10.2/lib/methsel2.g:250 called from
<function "HANDLE_METHOD_NOT_FOUND">( <arguments> )
 called from read-eval loop at *stdin*:200
type 'quit;' to quit to outer loop
brk> ImagesElm(f,AsList(Centre(D8))[1]);
RightCoset(<trivial group>,[ <identity ...>, <identity> of ... ])
brk> List(Centre(D8),x->ImageElm(f,x^2)*x^-1)
> ;
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method found for `*' on 2 arguments at /mnt/d/Download/gap-4.10.2/lib/methsel2.g:250 called from  entry * nonlist at /mnt/d/Download/gap-4.10.2/lib/tuples.gi:496 called from
func( C[elm] ) at /mnt/d/Download/gap-4.10.2/lib/coll.gi:743 called from
CallFuncList( ListOp, arg ) at /mnt/d/Download/gap-4.10.2/lib/coll.gi:677 called from
List( dpelm, function ( entry )
      return entry * nonlist;
  end ) at /mnt/d/Download/gap-4.10.2/lib/tuples.gi:496 called from
ImageElm( f, x ^ 2 ) * x ^ -1 at *errin*:2 called from
...  at *errin*:3
type 'quit;' to quit to outer loop
brk_2> quit;
brk> quitl
> quit;
Syntax warning: Unbound global variable in *errin*:4
quit;
^^^^
Error, Variable: 'quitl' must have a value in
  <corrupted statement>  called from
<function "HANDLE_METHOD_NOT_FOUND">( <arguments> )
 called from read-eval loop at *errin*:4
Syntax error: ; expected in *errin*:4
quit;
^^^^
brk> quit;
gap> List(Centre(D8),x->Image(f,x)*x^-1);
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method found for `*' on 2 arguments at /mnt/d/Download/gap-4.10.2/lib/methsel2.g:250 called from  entry * nonlist at /mnt/d/Download/gap-4.10.2/lib/tuples.gi:496 called from
func( C[elm] ) at /mnt/d/Download/gap-4.10.2/lib/coll.gi:743 called from
CallFuncList( ListOp, arg ) at /mnt/d/Download/gap-4.10.2/lib/coll.gi:677 called from
List( dpelm, function ( entry )
      return entry * nonlist;
  end ) at /mnt/d/Download/gap-4.10.2/lib/tuples.gi:496 called from
Image( f, x ) * x ^ -1 at *stdin*:200 called from
...  at *stdin*:200
type 'quit;' to quit to outer loop
brk> quit;
gap> x1=Centre(D8)[1];
Error, Variable: 'x1' must have a value
not in any function at *stdin*:200
gap> AsList(D8);
[ <identity ...>, a, a^-1, b, a^2, a*b, b*a, b*a^2 ]
gap> AsList(D8)[1];
<identity ...>
gap> Image(f, AsList(D8)[1]);
DirectProductElement( [ <identity ...>, <identity> of ... ] )
gap> List(Centre(D8),x->Image(f,x)*Image(Embedding(D8xC4,1),x)^-1);
[ DirectProductElement( [ <identity ...>, <identity> of ... ] ), DirectProductElement( [ a^2, f2 ] ) ]
gap> D:=AsGroup(List(Centre(D8),x->Image(f,x)*Image(Embedding(D8xC4,1),x)^-1));
<group of size 2 with 1 generators>
gap> FactorGroup(D8xC4,D);
Group([ f1, f2, f3, f4 ])
gap> StructureDescription(last);
"(C4 x C2) : C2"
gap>