  We compute the one-loop effective potential for the Polyakov loop on $S^3
times S^1$ for an asymptotically free gauge theory of arbitrary group $G$ and a
generic matter content. We apply this result to study the phase structures of
$G_2$, SO(N) and $Sp(2N)$ gauge theories which turn out to be in qualitative
agreement with the results of lattice calculations. On $S^3 \times S^1$, at
zero temperature, the Polyakov loop is zero for kinematical reasons. For small
but non-zero temperature, the Polyakov loop is still zero if the gauge theory
has an unbroken center, while it acquires a small vacuum expectation value for
gauge theories whose center is trivial or explicitly broken by the presence of
dynamical matter fields. At high temperatures, the saddle point structure of
the effective potential is different from the low temperature case suggesting
that the theory is in the deconfined phase. At finite $N$, the Polyakov loop is
non-zero in the high temperature phase only in theories with no unbroken center
symmetry, consistent with screening of the external charge introduced by the
Polyakov loop and Gauss law on a compact space.
