# Is there a general way to find the inverse of an automorphism of the free group? [closed]

If we describe an automorphism of the free group (on n generators) by where it sends the generators, is there some kind of algorithm to find the inverse automorphism? I am particularly interested in outer automorphisms.

Yes. Let $$X$$ be the given (ordered) free generating set of the free group, and let $$Y$$ be the set of images of the elements of $$X$$ under the automorphism.
Now perform Nielsen reduction on $$Y$$, and with each operation on $$Y$$, perform the same operation on $$X$$. Since $$Y$$ generates the group, Nielsen reduction will transform $$Y$$ back to $$X$$ and at the same time you are replacing $$X$$ by the set of inverse images of the the elements of $$X$$ under the automorphism.
Added later: Sorry, Nielsen reduction does not always work in general (although it seems to work on straightforward examples) and when it fails to reduce $$Y$$ to $$X$$ you have to use a more complicated set of reductions known as Whitehead automorphisms.
There is another method, which is more general and can be applied to an automorphism of any group $$G=\langle X \mid R \rangle$$ defined by a finite presentation that maps $$X$$ to $$Y$$. You carry out a modified version of the Todd-Coxeter coset enumeration on the subgroup $$H=\langle Y \rangle$$ of $$G$$. The index of the subgroup is of course $$1$$ and this should be successfully computed but, whereas basic coset enumeration just computes the coset table entries $$Hg_ix = Hg_j$$ for $$x \in X$$, the modified version also computes the corresponding $$h \in H$$ with $$g_ix = hg_j$$ as words over $$Y$$. So in this case, where there is only one coset, we end of with expressions for the elements $$x \in X$$ as words over $$Y$$, which is exactly what you are trying to compute.
• I particularly like how your description of this process mirrors the process for inverting a matrix: "Now perform row reduction on $Y$, and with each row operation on $Y$ perform the same operation on the identity matrix..." Commented Jul 15 at 23:13