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1 Orbital graphs

1.1 Constructing orbital graphs

1.1-1 OrbitalGraphs

Returns: A list of digraphs

This attribute is an immutable list of all orbital graphs of the permutation group G or transformation semigroup S.

The order of the returned list is not specified.

gap> OrbitalGraphs(DihedralGroup(IsPermGroup, 8));
[ <immutable digraph with 4 vertices, 4 edges>, 
  <immutable digraph with 4 vertices, 8 edges> ]

1.2 Values computed from orbital graphs

1.2-1 OrbitalClosure

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 (1.1-1). 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

1.2-2 OrbitalIndex

Returns: A positive integer

The orbital index of a permutation group is its Index (Ref 39.3-2) in its OrbitalClosure (1.2-1).

gap> OrbitalIndex(PSL(2,5));
12
gap> OrbitalIndex(PGL(2,5));
6
gap> OrbitalIndex(AlternatingGroup(6));
2
gap> OrbitalIndex(DihedralGroup(IsPermGroup, 6));
1

1.3 Recognising a group from its orbital graphs

1.3-1 IsOrbitalGraphRecognisable

Returns: true or false

A permutation group is orbital graph recognisable if and only if it is equal to its OrbitalClosure (1.2-1), i.e. if and only if its OrbitalIndex (1.2-2) is 1.

IsOGR is a synonym for IsOrbitalGraphRecognisable.

gap> IsOrbitalGraphRecognisable(QuaternionGroup(IsPermGroup, 8));
true
gap> IsOGR(AlternatingGroup(8));
false
gap> IsOGR(TrivialGroup(IsPermGroup));
true

1.3-2 IsStronglyOrbitalGraphRecognisable

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 (1.3-1).

IsStronglyOGR is a synonym for IsStronglyOrbitalGraphRecognisable.

gap> IsStronglyOrbitalGraphRecognisable(CyclicGroup(IsPermGroup, 8));
true
gap> IsStronglyOGR(QuaternionGroup(IsPermGroup, 8));
false
gap> IsStronglyOGR(TrivialGroup(IsPermGroup));
true

1.3-3 IsAbsolutelyOrbitalGraphRecognisable

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 (1.3-2).

IsAsolutelyOGR is a synonym for IsAbsolutelyOrbitalGraphRecognisable.

gap> IsAbsolutelyOrbitalGraphRecognisable(DihedralGroup(IsPermGroup, 8));
true
gap> IsAbsolutelyOGR(CyclicGroup(IsPermGroup, 8));
false
gap> IsAbsolutelyOGR(TrivialGroup(IsPermGroup));
true

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