‣ IsOrbitalGraph( D ) | ( category ) |
Returns: true or false
Every orbital graph that is constructed with the OrbitalGraphs package is a Digraphs package digraph (see IsDigraph (Digraphs 3.1-1)) that additionally lies in the category IsOrbitalGraph.
This makes it easy to recognise orbital graphs that were created with this package.
‣ IsOrbitalGraphOfGroup( D ) | ( category ) |
Returns: true or false
Every orbital graph that is constructed from a permutation group with the OrbitalGraphs package lies in the category IsOrbitalGraphOfGroup, which is a subcategory of IsOrbitalGraph (1.1-1).
‣ IsOrbitalGraphOfSemigroup( D ) | ( category ) |
Returns: true or false
Every orbital graph that is constructed from a transformation semigroup with the OrbitalGraphs package lies in the category IsOrbitalGraphOfGroup (1.1-2), which is a subcategory of IsOrbitalGraph (1.1-1).
‣ OrbitalGraphs( G ) | ( attribute ) |
‣ OrbitalGraphs( S ) | ( attribute ) |
‣ OrbitalGraphs( G, vertices ) | ( operation ) |
‣ OrbitalGraphs( G, max ) | ( operation ) |
Returns: A list of orbital graphs
This returns a list of all orbital graphs of the permutation group G or transformation semigroup S.
The order of the returned list is not specified.
gap> D8 := Group([ (1,2,3,4), (2,4) ]);; StructureDescription(D8); "D8" gap> OrbitalGraphs(D8); [ <self-paired orbital graph of D8 on 4 vertices with base-pair (1,3), 4 arcs> , <self-paired orbital graph of D8 on 4 vertices with base-pair (1,2), 8 arcs> ]
‣ OrbitalGraph( G, basepair, k ) | ( operation ) |
Returns: An orbital graph
If G is a permutation group, basepair is a pair of positive integers, and k is a positive integer such that [1..<A>k</A>] is preserved by G and contains the entries of basepair, then this function returns the orbital graph of G with the given basepair, on the vertices [1..<A>k</A>].
The resulting orbital graph will have basepair set as its BasePair (1.3-1) attribute.
gap> D8 := DihedralGroup(IsPermGroup, 8); Group([ (1,2,3,4), (2,4) ]) gap> OrbitalGraph(D8, [1, 3], 4); <self-paired orbital graph of Group([ (1,2,3,4), (2,4) ]) on 4 vertices with base-pair (1,3), 4 arcs> gap> OrbitalGraph(D8, [1, 3], 5); <self-paired orbital graph of Group([ (1,2,3,4), (2,4) ]) on 5 vertices with base-pair (1,3), 4 arcs> gap> G := Group([ (1,2)(3,4) ]);; gap> OrbitalGraph(G, [1, 2], 2); <self-paired orbital graph of Group([ (1,2)(3,4) ]) on 2 vertices with base-pair (1,2), 2 arcs>
‣ BasePair( D ) | ( attribute ) |
Returns: A list of two positive integers
If D is an orbital graph that was constructed with respect to a specific base pair, then this attribute stores that value.
Otherwise, is D is an orbital graph of a group, then this attribute stores the least edge of D, i.e. Minimum(DigraphEdges(<A>D</A>)); see DigraphEdges (Digraphs 5.1-3). If D is an orbital graph of a group, then this attribute stores an arbitrary base-pair of D.
Note that equal orbital graphs may have different base pairs, depending on how they were constructed.
gap> true; true
‣ UnderlyingGroup( D ) | ( attribute ) |
Returns: A permutation group
For an orbital graph D created from a permutation group G, this attribute stores the value G. Note that equal orbital graphs may have been created from different groups, and may therefore have different underlying groups.
gap> true; true
‣ UnderlyingSemigroup( D ) | ( attribute ) |
Returns: A transformation semigroup
For an orbital graph D created from a transformation semigroup S, this attribute stores the value S. Note that equal orbital graphs may have been created from different semigroups, and may therefore have different underlying semigroups.
gap> true; true
‣ OrbitalClosure( G ) | ( attribute ) |
Returns: A permutation group
The orbital closure of a nontrivial permutation group G is the intersection of the automorphism groups of all orbital graphs of the group. See OrbitalGraphs (1.2-1). A trivial permutation group is defined to be its own orbital closure.
For a transitive permutation group, OrbitalClosure returns the same as the GAP function TwoClosure (Ref 43.12-3) (which only applies to transitive groups).
gap> OrbitalClosure(PSL(2,5)) = SymmetricGroup(6); true gap> C6 := CyclicGroup(IsPermGroup, 6);; gap> OrbitalClosure(C6) = C6; true gap> A4_6 := Action(AlternatingGroup(4), Combinations([1..4], 2), OnSets);; gap> closure := OrbitalClosure(A4_6); Group([ (3,4), (2,5), (1,2,3)(4,6,5) ]) gap> IsConjugate(SymmetricGroup(6), > closure, WreathProduct(Group([(1,2)]), Group([(1,2,3)]))); true
‣ OrbitalIndex( G ) | ( attribute ) |
Returns: A positive integer
The orbital index of a permutation group is its Index (Ref 39.3-2) in its OrbitalClosure (1.4-1).
gap> OrbitalIndex(PSL(2,5)); 12 gap> OrbitalIndex(PGL(2,5)); 6 gap> OrbitalIndex(AlternatingGroup(6)); 2 gap> OrbitalIndex(DihedralGroup(IsPermGroup, 6)); 1
‣ IsOrbitalGraphRecognisable( G ) | ( property ) |
Returns: true or false
A permutation group is orbital graph recognisable if and only if it is equal to its OrbitalClosure (1.4-1), i.e. if and only if its OrbitalIndex (1.4-2) is 1.
IsOGR is a synonym for IsOrbitalGraphRecognisable.
gap> IsOrbitalGraphRecognisable(QuaternionGroup(IsPermGroup, 8)); true gap> IsOGR(AlternatingGroup(8)); false gap> IsOGR(TrivialGroup(IsPermGroup)); true
‣ IsStronglyOrbitalGraphRecognisable( G ) | ( property ) |
Returns: true or false
The nontrivial permutation group G is strongly orbital graph recognisable (strongly OGR) if and only if there exists some orbital graph of G whose automorphism group is G. The trivial permutation group is defined to be strongly OGR.
Note that every strongly OGR group is also orbital graph recognisable, see IsOrbitalGraphRecognisable (1.5-1).
IsStronglyOGR is a synonym for IsStronglyOrbitalGraphRecognisable.
gap> IsStronglyOrbitalGraphRecognisable(CyclicGroup(IsPermGroup, 8)); true gap> IsStronglyOGR(QuaternionGroup(IsPermGroup, 8)); false gap> IsStronglyOGR(TrivialGroup(IsPermGroup)); true
‣ IsAbsolutelyOrbitalGraphRecognisable( G ) | ( property ) |
Returns: true or false
The permutation group G is absolutely orbital graph recognisable (absolutely OGR) if and only if every orbital graph of G has automorphism group equal to G.
Note that every absolutely OGR group is also strongly orbital graph recognisable, see IsStronglyOrbitalGraphRecognisable (1.5-2).
IsAsolutelyOGR is a synonym for IsAbsolutelyOrbitalGraphRecognisable.
gap> IsAbsolutelyOrbitalGraphRecognisable(DihedralGroup(IsPermGroup, 8)); true gap> IsAbsolutelyOGR(CyclicGroup(IsPermGroup, 8)); false gap> IsAbsolutelyOGR(TrivialGroup(IsPermGroup)); true
‣ IsSelfPaired( arg ) | ( property ) |
Returns: true or false
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