The Conjugacy Problem in Wreath Products

Abstract

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 ${\hat{I}}^3,{\hat{I}}^2\in B$ the membership problem for $H{\hat{I}}^3⟨{\hat{I}}^2⟩$ is solvable; and (4) for any ${\hat{I}}^2\in B$ and any finite set of $n$ pairs of elements $(\hat{I}{\pm }_i,{\hat{I}}_i^3)$, where $\hat{I}{\pm }_i,{\hat{I}}_i^3\in B$ we can determine whether or not $ \big{[} \bigcap_{i=1}^n α^{-1}_iHγ_i \langle β\rangle\big{]} \cap C_B(β) = \emptyset $.

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