Example of a finite extension of an FAC-group not being an FAC-group. (Russian) Zbl 0601.20037

The group G is called conjugacy separable (FAC-group in the author’s terminology) if any two elements of G are conjugate in G if and only if their images are conjugate in every finite quotient of G. The author shows that the class of conjugacy separable groups is not closed under finite extensions. More precisely, let \[ G=(F_ k*(F_ n\times F_ m))\leftthreetimes ({\mathbb{Z}}_ 2[F_ n\times F_ m]) \] be a semidirect product, where \(F_ i\) is a free group of rank i, \(F_ k\) acts trivially and \(F_ n\times F_ m\) acts by left multiplications. Let H be a subgroup of \(F_ n\times F_ m\) with generators \(h_ 1,...,h_ k\), for which there is no algorithm whereby one can decide whether any element belongs to H. Let \(\phi\) be an automorphism of G of order 2, which acts trivially on \(F_ n\times F_ m\) and \(\phi (z_ j)=(z_ j,1-h_ j)\) for free generators \(z_ j\) of \(F_ k\). Let \(G^*=<\phi >\leftthreetimes G\) be a semidirect product. Then G is conjugacy separable, but \(G^*\) is not.
Reviewer: G.A.Noskov


20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20E26 Residual properties and generalizations; residually finite groups
20F05 Generators, relations, and presentations of groups
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
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