[MathJax on]

3 Faithful Transitive Permutation Representations

The action of a group G on the coset space of a subgroup gives us a transitive permutation representation of the group. Whenever the subgroup is core-free, we have that the action of G on the coset space of the subgroup will be faithful. Moreover, the stabilizer of a point on a faithful transitive permutation representation of G will always be a core-free subgroup.

3.1 Obtaining Faithful Transitive Permutation Representations

3.1-1 FaithfulTransitivePermutationRepresentations

Returns: a list

For a finite group G, FaithfulTransitivePermutationRepresentations returns a list of a faithful transitive permutation representation of G for each degree. If all_ftpr is true, then it will return a list of all faithful transitive permutation representations, up to conjugacy equivalence.

gap> sp := SymplecticGroup(4,2);;
gap> CoreFreeDegrees(sp);
[ 6, 10, 12, 15, 20, 30, 36, 40, 45, 60, 72, 80, 90, 120, 144, 180, 240, 360, 
  720 ]
gap> ftprs := FaithfulTransitivePermutationRepresentations(sp);; 
gap> Size(ftprs);
19
gap> all_ftprs := FaithfulTransitivePermutationRepresentations(sp,true);; 
gap> Size(all_ftprs);
54

3.2 Faithful Transitive Permutation Representation of Minimal Degree

To complement the already existing functions in GAP MinimalFaithfulPermutationDegree and MinimalFaithfulPermutationRepresentation, the following functions to retrieve the MinimalFaithfulTransitivePermutationRepresentation and MinimalFaithfulTransitivePermutationDegree.

3.2-1 MinimalFaithfulTransitivePermutationRepresentation

Returns: an isomorphism (or a list of isomorphisms)

For a finite group G, MinimalFaithfulTransitivePermutationRepresentation returns an isomorphism of G into the symmetric group of minimal degree acting transitively on its domain. If all_minimal_ftpr is set as true, then it returns a list of all isomorphisms G into the symmetric group of minimal degree.

gap> sp := SymplecticGroup(4,2);;
gap> min_ftpr := MinimalFaithfulTransitivePermutationRepresentation(sp);
CompositionMapping( <action epimorphism>, <action isomorphism> )
gap> min_ftprs := MinimalFaithfulTransitivePermutationRepresentation(sp,true);
[ CompositionMapping( <action epimorphism>, <action isomorphism> ), 
  CompositionMapping( <action epimorphism>, <action isomorphism> ) ]

3.2-2 MinimalFaithfulTransitivePermutationDegree

Returns: an integer

For a finite group G, MinimalFaithfulTransitivePermutationDegree returns the least positive integer n such that G is isomorphic to a subgroup of the symmetric group of degree n acting transitively on its domain.

gap> sp := SymplecticGroup(4,2);; g:=SimpleGroup("PSL",3,5);;
gap> MinimalFaithfulTransitivePermutationDegree(sp);
6
gap> MinimalFaithfulTransitivePermutationDegree(g);
31

3.3 Faithful Transitive Permutation Representation of given Degree

To obtain a faithful transitive permutation Representation of a specific degree, the following function FaithfulTransitivePermutationRepresentationsOfDegree can be used.

3.3-1 FaithfulTransitivePermutationRepresentationsOfDegree

Returns: an isomorphism (or a list of isomorphisms)

For a finite group G, FaithfulTransitivePermutationRepresentationsOfDegree returns one isomorphism of G into the symmetric group of degree d acting transitively on its domain. If all_ftpr_of_given_degree is set as true, then it returns a list of all isomorphisms G into the symmetric group of degree d, up to conjugacy equivalence.

gap> sp := SymplecticGroup(4,2);;
gap> FaithfulTransitivePermutationRepresentationsOfDegree(sp,10);
CompositionMapping( <action epimorphism>, <action isomorphism> )
gap> FaithfulTransitivePermutationRepresentationsOfDegree(sp,20, true);
[ CompositionMapping( <action epimorphism>, <action isomorphism> ), 
  CompositionMapping( <action epimorphism>, <action isomorphism> ), 
  CompositionMapping( <action epimorphism>, <action isomorphism> ) ]

generated by GAPDoc2HTML