The Conjugacy Problem in Wreath Products

In 1966 Jane Matthews claimed that the conjugacy problem is solvable in the standard restricted wreath product $A \wr B$ of two nontrivial groups $A$ and $B$ if and only if (i) the conjugacy problem is solvable in $A$ and $B$ and (ii) $B$ has a \textit{solvable power problem}. We show that there should be an additional condition that either $A$ is abelian or $B$ has a \textit{solvable order problem}. We also show that, if $A$ and $B$ are non-trivial recursively presented groups where $A$ has an infinite number of conjugacy classes and $B$ acts on $B/H$ transitively, then the conjugacy problem in the permutational restricted wreath product $A \wr_{B/H} B$ is solvable if and only if the following hold: (1) the conjugacy problem is solvable in $A$ and in $B$; (2) either $A$ is abelian or \textit{the orbit order problem} is solvable in $B$; (3) for any $\gamma, \beta \in B$ the membership problem for $H \gamma \langle \beta \rangle$ is solvable; and (4) for any $\beta \in B$ and any finite set of $n$ pairs of elements $(\alpha_i, \gamma_i)$, where $\alpha_i, \gamma_i \in B$ we can determine whether or not \big{[} \bigcap_{i=1}^n \alpha^{-1}_iH\gamma_i \langle \beta \rangle\big{]} \cap C_B(\beta) = \emptyset .