Group Induced Isomorphism (GII)

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

Does the mapping $\varphi$ defined by $\varphi(u_i) = v_i$ for $i = 1, \ldots, m$, induce an isomorphism from $\langle U \rangle$ to $\langle V \rangle$?

Status

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

Remarks

A variant of the Nielsen reduction algorithm can be used to show the problem is in $\P$. The reduction to show hardness is from GI.