Automorphisms with finite Reidemeister number in residually finite groups.
Let \(f\) be an automorphism of the group \(G\) of finite order. Say elements \(x\) and \(y\) of \(G\) are related if there exists \(g\) in \(G\) with \(gx=yg^f\). This defines an equivalence relation on \(G\). Assume also that its number of equivalence classes is finite. The author proves the following results. If \(f\) has prime order and if either \(G\) is residually finite or \(G\) is soluble and finitely generated, then \(G\) is nilpotent-by-finite. If \(G\) is linear and finitely generated, then \(G\) is soluble-by-finite. In the first result the prime restriction on \(f\) cannot be removed; the author gives finitely generated, metabelian counterexamples with \(f\) of order 4 and just 2 equivalence classes. The paper also contains a brief survey of where automorphisms (even endomorphisms) of this type have arisen previously in the literature.
scientific article; zbMATH DE number 5380145
scientific article