  Let $k$ be a perfect field such that for every $n$ there are only finitely
many field extensions, up to isomorphism, of $k$ of degree $n$. If $G$ is a
reductive algebraic group defined over $k$, whose characteristic is very good
for $G$, then we prove that $G(k)$ has only finitely many $z$-classes.
  For each perfect field $k$ which does not have the above finiteness property
we show that there exist groups $G$ over $k$ such that $G(k)$ has infinitely
many $z$-classes.
