  We study the Polyakov loop and the $Z_2$ symmetry in the lattice $Z_2+$Higgs
theory in 4D Euclidean space using Monte Carlo simulations. The results show
that this symmetry is realised in the Higgs symmetric phase for large number of
temporal lattice sites. To understand the dependence on the number of temporal
sites, we consider a one dimensional model by keeping terms of the original
action corresponding to a single spatial site. In this approximation the
partition function can be calculated exactly as a function of the Polyakov
loop. The resulting free energy is found to have the $Z_2$ symmetry in the
limit of large temporal sites. We argue that this is due to $Z_2$ invariance as
well as dominance of the distribution or density of states corresponding to the
action.
