The orbits of the \((\varphi,\psi)\)-twisted conjugacy action are called the \((\varphi,\psi)\)-twisted conjugacy classes or the Reidemeister classes of \((\varphi,\psi)\). We denote the twisted conjugacy class of \(g \in G\) by \([g]_{\varphi,\psi}\).
‣ TwistedConjugacyClass( hom1[, hom2], g ) | ( function ) |
‣ ReidemeisterClass( hom1[, hom2], g ) | ( function ) |
Returns: the (hom1,hom2)-twisted conjugacy class of g.
gap> tcc := TwistedConjugacyClass( phi, psi, g1 ); (4,6,5)^G
‣ Representative( tcc ) | ( attribute ) |
Returns: the group element that was used to construct tcc.
‣ ActingDomain( tcc ) | ( attribute ) |
Returns: the group whose twisted conjugacy action tcc is an orbit of.
‣ FunctionAction( tcc ) | ( attribute ) |
Returns: the twisted conjugacy action that tcc is an orbit of.
‣ \in( g, tcc ) | ( operation ) |
Returns: true if g is an element of tcc, otherwise false.
‣ IsFinite( tcc ) | ( property ) |
Returns: true if tcc is finite, otherwise false.
‣ Size( tcc ) | ( attribute ) |
Returns: the number of elements in tcc.
This is calculated using the orbit-stabiliser theorem.
‣ StabiliserOfExternalSet( tcc ) | ( attribute ) |
‣ StabilizerOfExternalSet( tcc ) | ( attribute ) |
Returns: the stabiliser of Representative(tcc) under the action FunctionAction(tcc).
‣ List( tcc ) | ( function ) |
‣ AsList( tcc ) | ( attribute ) |
Returns: a list containing the elements of tcc.
If tcc is infinite, this will run forever or cause an error. It is recommended to first test the finiteness of tcc using IsFinite (4.2-5).
‣ Random( tcc ) | ( operation ) |
Returns: a random element in tcc.
‣ \=( tcc1, tcc2 ) | ( operation ) |
Returns: true if tcc1 is equal to tcc2, otherwise false.
gap> Representative( tcc ); (4,6,5) gap> ActingDomain( tcc ) = H; true gap> FunctionAction( tcc )( g1, h2 ); (1,6,4,2)(3,5) gap> List( tcc ); [ (4,6,5), (1,6,4,2)(3,5) ] gap> Size( tcc ); 2 gap> StabiliserOfExternalSet( tcc ); Group([ (1,2,3,4,5), (1,3,4,5,2) ])
‣ TwistedConjugacyClasses( hom1[, hom2][, N] ) | ( function ) |
‣ ReidemeisterClasses( hom1[, hom2][, N] ) | ( function ) |
Returns: a list containing the (hom1, hom2)-twisted conjugacy classes if there are finitely many, or fail otherwise.
If the argument N is provided, it must be a normal subgroup of Range(hom1); the function will then only return the Reidemeister classes that intersect N non-trivially. It is guaranteed that the Reidemeister class of the identity is in the first position, and that the representatives of the classes belong to N if this argument is provided.
If \(G\) and \(H\) are finite, this function relies on an orbit-stabiliser algorithm. Otherwise, it relies on the algorithms in [DT21] and [Ter25].
‣ RepresentativesTwistedConjugacyClasses( hom1[, hom2][, N] ) | ( function ) |
‣ RepresentativesReidemeisterClasses( hom1[, hom2][, N] ) | ( function ) |
Returns: a list containing representatives of the (hom1, hom2)-twisted conjugacy classes if there are finitely many, or fail otherwise.
If the argument N is provided, it must be a normal subgroup of Range(hom1); the function will then only return the representatives of the twisted conjugacy classes that intersect N non-trivially. It is guaranteed that the identity is in the first position, and that all elements belong to N if this argument is provided.
gap> TwistedConjugacyClasses( phi, psi ){ [ 1 .. 7 ] }; [ ()^G, (4,5,6)^G, (4,6,5)^G, (3,4)(5,6)^G, (3,4,5)^G, (3,4,6)^G, (3,5,4)^G ] gap> RepresentativesTwistedConjugacyClasses( phi, psi ){ [ 1 .. 7 ] }; [ (), (4,5,6), (4,6,5), (3,4)(5,6), (3,4,5), (3,4,6), (3,5,4) ]
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