Conjugate Subgroups (CS)

Input

Let $S$ be a finite set and $F$ be the free group generated by $S$. Let $\overline{S} = \{s, s^{-1} \mid s \in S\}$, where $s^{-1}$ denotes the inverse of $s$. Let $\overline{S}^\ast$ denote the set of all finite words over $\overline{S}$. Let $U = \{u_1, \ldots, u_m\}$, $V = \{v_1, \ldots, v_p\} \subseteq \overline{S}^\ast$, where $m = p \in \N$.

Problem

Is there a $x \in \overline{S}^\ast$ such that $x^{-1} \langle U \rangle x = \langle V \rangle$?

Status

$\P$-complete (Avenhaus and Madlener [1]).

Remarks

A polynomial time algorithm for the problem is given in [1]. The reduction to show hardness is from Equivalence of Right Cosets, see IC.

The problem of determining whether $x^{-1} \langle U \rangle x$ is a subgroup of $\langle V \rangle$ is also $\P$-complete [1].