Normal closure of finite subgroups of $\mathrm{Aut}(F_n)$ and $\mathrm{Out}(F_n)$

Abstract

For $n\ge 3$, let $G$ be a nontrivial finite subgroup of $\mathrm{Aut}(F_n)$ with $|G|$ not a power of $2$. We prove that the normal closure $N(G)$ is $\mathrm{SAut}(F_n)$ if $G\subset \mathrm{SAut}(F_n)$ and $N(G)$ is $\mathrm{Aut}(F_n)$ otherwise. When $|G|$ is a power of $2$, we have a partial theorem. Similarly, let $G^{\prime }$ be a nontrivial finite subgroup of $\mathrm{Out}(F_n)$ with $|G^{\prime }|$ not a power of $2$. Then the normal closure $N(G^{\prime })$ is $\mathrm{SOut}(F_n)$ if $G^{\prime }\subset \mathrm{SOut}(F_n)$ and $N(G^{\prime })$ is $\mathrm{Out}(F_n)$ otherwise. When $|G^{\prime }|$ is a power of $2$, we have a partial theorem as well.