A positive integer $n$ is said to be a finitistic order for an element $u$ of a group $F$ if there exist a finite group $G$ and a homomorphism $h$ from $F$ into $G$ such that $h(u)$ has order $n$ (in $G$). Let $u$ be a non-identity element of a free group $F$. It is well known that $u$ has infinite order. We prove that any positive integer is a finitistic order for $u$.
This is a joint work with Conceição Nogueira and M. Lurdes Teixeira.