In the book Lyndon, Schupp, Combinatorial Group Theory, P.30 in the edition from 2000 They mention an unpublished work by Waldhausen that is said to give an algorithm to determine whether two subgroups are automorphic given their free generators. I searched for papers written by Waldhausen But I didn't find it. Has Waldhausen or anyone else published a solution to this problem? Where can i find it? This is a copy of a post from [StackExchange](https://math.stackexchange.com/questions/2677657/an-algorithm-determining-whether-two-subgroups-of-a-free-group-are-automorphic) 



An algorithm determining whether two subgroups of a finitely generated free group are automorphic
According to a comment on the math.SE link, the OP wants to understand the question of deciding when two (finitely generated) subgroups of a given free group $F$ are equivalent by an automorphism of $F$.


I believe that this was first solved by Gersten:



> 
> S. Gersten, On Whitehead’s algorithm, Bull. Am. Math. Soc. 10 (1984) 281–
>  284.
> 
> 
> Abstract. One can decide effectively when two finitely generated subgroups of a finitely generated free group $F$ are equivalent under an automorphism of $F$. The subgroup of automorphisms of $F$ mapping a given finitely generated subgroup $S$ of $F$ into a conjugate of $S$ is finitely
>  presented.
> 
> 
> 


Gersten states that the question asked by the OP was first raised by Whitehead. He also states that he settles this question of Whitehead, and does not cite any papers of Waldhausen. So I guess the paper of Waldhausen never appeared...


