The number of twisted conjugacy classes is called the Reidemeister number and is always a positive integer or infinity.
‣ ReidemeisterNumber( hom1[, hom2] ) | ( function ) |
‣ NrTwistedConjugacyClasses( hom1[, hom2] ) | ( function ) |
Returns: the Reidemeister number of ( hom1, hom2 ).
If G is abelian, this function relies on (a generalisation of) [Jia83, Thm. 2.5]. If G = H, G is finite non-abelian and \psi = \operatorname{id}_G, it relies on [FH94, Thm. 5]. Otherwise, it simply calculates the twisted conjugacy classes and then counts them.
gap> Q := QuaternionGroup( 8 );; gap> phi := GroupHomomorphismByImages( Q, Q, [ Q.1, Q.2 ], [ Q.2, Q.1 ] );; gap> ReidemeisterNumber( phi ); 3 gap> D := DihedralGroup( 8 );; gap> psi := GroupHomomorphismByImages( Q, D, [ Q.1, Q.2 ], [ D.1 * D.2, D.3 ] );; gap> chi := GroupHomomorphismByImages( Q, D, [ Q.1, Q.2 ], [ D.1, D.3 ] );; gap> ReidemeisterNumber( psi, chi ); 4
The set of all Reidemeister numbers of automorphisms is called the Reidemeister spectrum and is denoted by \operatorname{Spec}_R(G), i.e.
\operatorname{Spec}_R(G) := \{\, R(\varphi) \mid \varphi \in \operatorname{Aut}(G) \,\}.
The set of all Reidemeister numbers of endomorphisms is called the extended Reidemeister spectrum and is denoted by \operatorname{ESpec}_R(G), i.e.
\operatorname{ESpec}_R(G) := \{\, R(\varphi) \mid \varphi \in \operatorname{End}(G) \,\}.
The set of all Reidemeister numbers of pairs of homomorphisms from a group H to a group G is called the coincidence Reidemeister spectrum of H and G and is denoted by \operatorname{CSpec}_R(H,G), i.e.
\operatorname{CSpec}_R(H,G) := \{\, R(\varphi, \psi) \mid \varphi,\psi \in \operatorname{Hom}(H,G) \,\}.
If H = G this is also denoted by \operatorname{CSpec}_R(G). The set of all Reidemeister numbers of pairs of homomorphisms from every group H to a group G is called the total Reidemeister spectrum and is denoted by \operatorname{TSpec}_R(G), i.e.
\operatorname{TSpec}_R(G) := \bigcup_{H} \operatorname{CSpec}_R(H,G).
Please note that the functions below are only implemented for finite groups.
‣ ReidemeisterSpectrum( G ) | ( function ) |
Returns: the Reidemeister spectrum of G.
If G is abelian, this function relies on the results from [Sen23]. Otherwise, it relies on [FH94, Thm. 5].
‣ ExtendedReidemeisterSpectrum( G ) | ( function ) |
Returns: the extended Reidemeister spectrum of G.
If G is abelian, this is just the set of all divisors of the order of G. Otherwise, this function relies on [FH94, Thm. 5].
‣ CoincidenceReidemeisterSpectrum( [H, ]G ) | ( function ) |
Returns: the coincidence Reidemeister spectrum of H and G.
‣ TotalReidemeisterSpectrum( G ) | ( function ) |
Returns: the total Reidemeister spectrum of H and G.
gap> ReidemeisterSpectrum( Q ); [ 2, 3, 5 ] gap> ExtendedReidemeisterSpectrum( Q ); [ 1, 2, 3, 5 ] gap> CoincidenceReidemeisterSpectrum( Q ); [ 1, 2, 3, 4, 5, 8 ] gap> CoincidenceReidemeisterSpectrum( D, Q ); [ 4, 8 ] gap> CoincidenceReidemeisterSpectrum( Q, D ); [ 2, 3, 4, 6, 8 ] gap> TotalReidemeisterSpectrum( Q ); [ 1, 2, 3, 4, 5, 6, 8 ]
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