  
  [1X1 [33X[0;0YOrbital graphs[133X[101X
  
  
  [1X1.1 [33X[0;0YCategories of orbital graphs[133X[101X
  
  [1X1.1-1 IsOrbitalGraph[101X
  
  [33X[1;0Y[29X[2XIsOrbitalGraph[102X( [3XD[103X ) [32X Category[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X[133X
  
  [33X[0;0YEvery  orbital graph that is constructed with the [5XOrbitalGraphs[105X package is a
  [5XDigraphs[105X   package   digraph  (see  [2XIsDigraph[102X  ([14XDigraphs:  IsDigraph[114X))  that
  additionally lies in the category [2XIsOrbitalGraph[102X.[133X
  
  [33X[0;0YThis  makes  it easy to recognise orbital graphs that were created with this
  package.[133X
  
  [1X1.1-2 IsOrbitalGraphOfGroup[101X
  
  [33X[1;0Y[29X[2XIsOrbitalGraphOfGroup[102X( [3XD[103X ) [32X Category[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X[133X
  
  [33X[0;0YEvery  orbital  graph  that is constructed from a permutation group with the
  [5XOrbitalGraphs[105X package lies in the category [2XIsOrbitalGraphOfGroup[102X, which is a
  subcategory of [2XIsOrbitalGraph[102X ([14X1.1-1[114X).[133X
  
  [1X1.1-3 IsOrbitalGraphOfSemigroup[101X
  
  [33X[1;0Y[29X[2XIsOrbitalGraphOfSemigroup[102X( [3XD[103X ) [32X Category[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X[133X
  
  [33X[0;0YEvery orbital graph that is constructed from a transformation semigroup with
  the   [5XOrbitalGraphs[105X  package  lies  in  the  category  [2XIsOrbitalGraphOfGroup[102X
  ([14X1.1-2[114X), which is a subcategory of [2XIsOrbitalGraph[102X ([14X1.1-1[114X).[133X
  
  
  [1X1.2 [33X[0;0YConstructing orbital graphs[133X[101X
  
  [1X1.2-1 OrbitalGraphs[101X
  
  [33X[1;0Y[29X[2XOrbitalGraphs[102X( [3XG[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XOrbitalGraphs[102X( [3XS[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XOrbitalGraphs[102X( [3XG[103X, [3Xvertices[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XOrbitalGraphs[102X( [3XG[103X, [3Xmax[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA list of orbital graphs[133X
  
  [33X[0;0YThis  returns  a  list  of  all orbital graphs of the permutation group [3XG[103X or
  transformation semigroup [3XS[103X.[133X
  
  [33X[0;0YThe order of the returned list is not specified.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD8 := Group([ (1,2,3,4), (2,4) ]);; StructureDescription(D8);[127X[104X
    [4X[28X"D8"[128X[104X
    [4X[25Xgap>[125X [27XOrbitalGraphs(D8);[127X[104X
    [4X[28X[ <self-paired orbital graph of D8 on 4 vertices with base-pair (1,3), 4 arcs>[128X[104X
    [4X[28X    , <self-paired orbital graph of D8 on 4 vertices [128X[104X
    [4X[28X    with base-pair (1,2), 8 arcs> ][128X[104X
  [4X[32X[104X
  
  [1X1.2-2 OrbitalGraph[101X
  
  [33X[1;0Y[29X[2XOrbitalGraph[102X( [3XG[103X, [3Xbasepair[103X, [3Xk[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YAn orbital graph[133X
  
  [33X[0;0YIf  [3XG[103X is a permutation group, [3Xbasepair[103X is a pair of positive integers, and [10Xk[110X
  is a positive integer such that [10X[1..<A>k</A>][110X is preserved by [3XG[103X and contains
  the  entries  of [3Xbasepair[103X, then this function returns the orbital graph of [3XG[103X
  with the given [3Xbasepair[103X, on the vertices [10X[1..<A>k</A>][110X.[133X
  
  [33X[0;0YThe  resulting  orbital graph will have [3Xbasepair[103X set as its [2XBasePair[102X ([14X1.3-1[114X)
  attribute.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD8 := DihedralGroup(IsPermGroup, 8);[127X[104X
    [4X[28XGroup([ (1,2,3,4), (2,4) ])[128X[104X
    [4X[25Xgap>[125X [27XOrbitalGraph(D8, [1, 3], 4);[127X[104X
    [4X[28X<self-paired orbital graph of Group([ (1,2,3,4), (2,4) ]) on 4 vertices [128X[104X
    [4X[28Xwith base-pair (1,3), 4 arcs>[128X[104X
    [4X[25Xgap>[125X [27XOrbitalGraph(D8, [1, 3], 5);[127X[104X
    [4X[28X<self-paired orbital graph of Group([ (1,2,3,4), (2,4) ]) on 5 vertices [128X[104X
    [4X[28Xwith base-pair (1,3), 4 arcs>[128X[104X
    [4X[25Xgap>[125X [27XG := Group([ (1,2)(3,4) ]);;[127X[104X
    [4X[25Xgap>[125X [27XOrbitalGraph(G, [1, 2], 2);[127X[104X
    [4X[28X<self-paired orbital graph of Group([ (1,2)(3,4) ]) on 2 vertices [128X[104X
    [4X[28Xwith base-pair (1,2), 2 arcs>[128X[104X
  [4X[32X[104X
  
  
  [1X1.3 [33X[0;0YInformation stored about orbital graphs at creation[133X[101X
  
  [1X1.3-1 BasePair[101X
  
  [33X[1;0Y[29X[2XBasePair[102X( [3XD[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA list of two positive integers[133X
  
  [33X[0;0YIf  [3XD[103X  is  an  orbital graph that was constructed with respect to a specific
  base pair, then this attribute stores that value.[133X
  
  [33X[0;0YOtherwise,  is  [3XD[103X is an orbital graph of a group, then this attribute stores
  the  least edge of [3XD[103X, i.e. [10XMinimum(DigraphEdges(<A>D</A>))[110X; see [2XDigraphEdges[102X
  ([14XDigraphs:  DigraphEdges[114X).  If  [3XD[103X  is an orbital graph of a group, then this
  attribute stores an arbitrary base-pair of [3XD[103X.[133X
  
  [33X[0;0YNote  that  equal orbital graphs may have different base pairs, depending on
  how they were constructed.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtrue;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X1.3-2 UnderlyingGroup[101X
  
  [33X[1;0Y[29X[2XUnderlyingGroup[102X( [3XD[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA permutation group[133X
  
  [33X[0;0YFor  an  orbital  graph [3XD[103X created from a permutation group [10XG[110X, this attribute
  stores  the  value  [10XG[110X.  Note that equal orbital graphs may have been created
  from different groups, and may therefore have different underlying groups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtrue;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X1.3-3 UnderlyingSemigroup[101X
  
  [33X[1;0Y[29X[2XUnderlyingSemigroup[102X( [3XD[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA transformation semigroup[133X
  
  [33X[0;0YFor  an  orbital  graph  [3XD[103X  created  from a transformation semigroup [10XS[110X, this
  attribute  stores  the value [10XS[110X. Note that equal orbital graphs may have been
  created   from  different  semigroups,  and  may  therefore  have  different
  underlying semigroups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtrue;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X1.4 [33X[0;0YValues computed from the orbital graphs of a group[133X[101X
  
  [1X1.4-1 OrbitalClosure[101X
  
  [33X[1;0Y[29X[2XOrbitalClosure[102X( [3XG[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA permutation group[133X
  
  [33X[0;0YThe  [13Xorbital closure[113X of a nontrivial permutation group [3XG[103X is the intersection
  of  the  automorphism  groups  of  all  orbital  graphs  of  the  group. See
  [2XOrbitalGraphs[102X  ([14X1.2-1[114X). A trivial permutation group is defined to be its own
  orbital closure.[133X
  
  [33X[0;0YFor  a  transitive permutation group, [2XOrbitalClosure[102X returns the same as the
  [5XGAP[105X  function  [2XTwoClosure[102X  ([14XReference:  TwoClosure[114X)  (which  only applies to
  transitive groups).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XOrbitalClosure(PSL(2,5)) = SymmetricGroup(6);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XC6 := CyclicGroup(IsPermGroup, 6);;[127X[104X
    [4X[25Xgap>[125X [27XOrbitalClosure(C6) = C6;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XA4_6 := Action(AlternatingGroup(4), Combinations([1..4], 2), OnSets);;[127X[104X
    [4X[25Xgap>[125X [27Xclosure := OrbitalClosure(A4_6);[127X[104X
    [4X[28XGroup([ (3,4), (2,5), (1,2,3)(4,6,5) ])[128X[104X
    [4X[25Xgap>[125X [27XIsConjugate(SymmetricGroup(6),[127X[104X
    [4X[25X>[125X [27X               closure, WreathProduct(Group([(1,2)]), Group([(1,2,3)])));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X1.4-2 OrbitalIndex[101X
  
  [33X[1;0Y[29X[2XOrbitalIndex[102X( [3XG[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA positive integer[133X
  
  [33X[0;0YThe  [13Xorbital index[113X of a permutation group is its [2XIndex[102X ([14XReference: Index for
  a group and its subgroup[114X) in its [2XOrbitalClosure[102X ([14X1.4-1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XOrbitalIndex(PSL(2,5));[127X[104X
    [4X[28X12[128X[104X
    [4X[25Xgap>[125X [27XOrbitalIndex(PGL(2,5));[127X[104X
    [4X[28X6[128X[104X
    [4X[25Xgap>[125X [27XOrbitalIndex(AlternatingGroup(6));[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XOrbitalIndex(DihedralGroup(IsPermGroup, 6));[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  
  [1X1.5 [33X[0;0YRecognising a group from its orbital graphs[133X[101X
  
  [1X1.5-1 IsOrbitalGraphRecognisable[101X
  
  [33X[1;0Y[29X[2XIsOrbitalGraphRecognisable[102X( [3XG[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X[133X
  
  [33X[0;0YA permutation group is [13Xorbital graph recognisable[113X if and only if it is equal
  to  its [2XOrbitalClosure[102X ([14X1.4-1[114X), i.e. if and only if its [2XOrbitalIndex[102X ([14X1.4-2[114X)
  is [10X1[110X.[133X
  
  [33X[0;0Y[10XIsOGR[110X is a synonym for [2XIsOrbitalGraphRecognisable[102X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsOrbitalGraphRecognisable(QuaternionGroup(IsPermGroup, 8));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsOGR(AlternatingGroup(8));[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsOGR(TrivialGroup(IsPermGroup));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X1.5-2 IsStronglyOrbitalGraphRecognisable[101X
  
  [33X[1;0Y[29X[2XIsStronglyOrbitalGraphRecognisable[102X( [3XG[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X[133X
  
  [33X[0;0YThe  nontrivial  permutation  group [3XG[103X is [13Xstrongly orbital graph recognisable
  (strongly  OGR)[113X  if  and  only if there exists [13Xsome[113X orbital graph of [3XG[103X whose
  automorphism  group  is  [3XG[103X.  The  trivial permutation group is defined to be
  strongly OGR.[133X
  
  [33X[0;0YNote  that  every strongly OGR group is also orbital graph recognisable, see
  [2XIsOrbitalGraphRecognisable[102X ([14X1.5-1[114X).[133X
  
  [33X[0;0Y[10XIsStronglyOGR[110X is a synonym for [2XIsStronglyOrbitalGraphRecognisable[102X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsStronglyOrbitalGraphRecognisable(CyclicGroup(IsPermGroup, 8));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsStronglyOGR(QuaternionGroup(IsPermGroup, 8));[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsStronglyOGR(TrivialGroup(IsPermGroup));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X1.5-3 IsAbsolutelyOrbitalGraphRecognisable[101X
  
  [33X[1;0Y[29X[2XIsAbsolutelyOrbitalGraphRecognisable[102X( [3XG[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X[133X
  
  [33X[0;0YThe permutation group [3XG[103X is [13Xabsolutely orbital graph recognisable (absolutely
  OGR)[113X if and only if [13Xevery[113X orbital graph of [3XG[103X has automorphism group equal to
  [3XG[103X.[133X
  
  [33X[0;0YNote  that  every  absolutely  OGR  group  is  also  strongly  orbital graph
  recognisable, see [2XIsStronglyOrbitalGraphRecognisable[102X ([14X1.5-2[114X).[133X
  
  [33X[0;0Y[10XIsAsolutelyOGR[110X is a synonym for [2XIsAbsolutelyOrbitalGraphRecognisable[102X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsAbsolutelyOrbitalGraphRecognisable(DihedralGroup(IsPermGroup, 8));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsAbsolutelyOGR(CyclicGroup(IsPermGroup, 8));[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsAbsolutelyOGR(TrivialGroup(IsPermGroup));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X1.6 [33X[0;0YAttributes and properties of individual orbital graphs[133X[101X
  
  [1X1.6-1 IsSelfPaired[101X
  
  [33X[1;0Y[29X[2XIsSelfPaired[102X( [3Xarg[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X[133X
  
