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