  In this paper, we study prime order automorphisms of generalized quadrangles.
We show that, if $\mathcal{Q}$ is a thick generalized quadrangle of order
$(s,t)$, where $s > t$ and $s+1$ is prime, and $\mathcal{Q}$ has an
automorphism of order $s+1$, then \[ s \left\lceil \left\lceil
\frac{t^2}{s+1}\right\rceil\left(\frac{s+1}{t} \right) \right\rceil \le
t(s+t),\] with a similar inequality holding in the dual case when $t > s$,
$t+1$ is prime, and $\mathcal{Q}$ is a thick generalized quadrangle of order
$(s,t)$ with an automorphism of order $t+1$.
  In particular, if $s+1$ is prime and if there exists a natural number $n$
such that \[ \frac{t^2}{n+1} + t \le s + 1 < \frac{t^2}{n},\] then a thick
generalized quadrangle $\mathcal{Q}$ cannot have an automorphism of order
$s+1$, and hence the automorphism group of $\mathcal{Q}$ cannot be transitive
on points. These results apply to numerous potential orders for which it is
still unknown whether or not generalized quadrangles exist, showing that any
examples would necessarily be somewhat asymmetric. Finally, we are able to use
the theory we have built up about prime order automorphisms of generalized
quadrangles to show that the automorphism group of a potential generalized
quadrangle of order $(4,12)$ must necessarily be intransitive on both points
and lines.
