  Let $G=\ast_{i=1}^{n}G_{i}$ and let $\phi$ be a symmetric endomorphism of
$G$. If $\phi$ is a monomorphism or if $G$ is a finitely generated residually
finite group, then the fixed subgroup $Fix(\phi)=\{g\in G:\phi(g)=g\}$ of
$\phi$ has Kurosh rank at most $n$.
