Let G, H be groups and \varphi,\psi\colon H \to G group homomorphisms. Then the pair (\varphi,\psi) induces a (right) group action on G given by
G \times H \to G\colon (g,h) \mapsto g \cdot h = \psi(h)^{-1} g\varphi(h).
This group action is called (\varphi,\psi)-twisted conjugation, and induces an equivalence relation \sim_{\varphi,\psi} on G:
g_1 \sim_{\varphi,\psi} g_2 \iff \exists h \in H: g_1 \cdot h = g2.
The equivalence classes (i.e. the orbits of the action) are called Reidemeister classes and the number of Reidemeister classes is called the Reidemeister number R(\varphi,\psi) of the pair (\varphi,\psi). The stabiliser of the identity 1_G for this action is the coincidence group \operatorname{Coin}(\varphi, \psi ), i.e. the subgroup of H given by
\operatorname{Coin}(\varphi,\psi) := \{\, h \in H \mid \varphi(h) = \psi(h) \,\}.
The TwistedConjugacy package provides methods to calculate Reidemeister classes, Reidemeister numbers and coincidence groups of pairs of group homomorphisms. These methods are implemented for finite groups and polycyclically presented groups. If H and G are both infinite polycyclically presented groups, then some methods in this package are only guaranteed to produce a result if either G = H or G is nilpotent-by-finite. Otherwise, these methods may potentially throw an error: "Error, no method found!"
Bugs in this package, in GAP or any other package used directly or indirectly, may cause functions from this package to produce errors or even wrong results. You can set the variable ASSERT@TwistedConjugacy to true, which will cause certain functions to verify the correctness of their output. This should make results more (but not completely!) reliable, at the cost of some performance.
When using this package with PcpGroups, you can do the same for Polycyclic's variables CHECK_CENT@Polycyclic, CHECK_IGS@Polycyclic and CHECK_INTSTAB@Polycyclic.
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