Z_N-symmetry in gauge theories at finite temperatures

The role of Z_N symmetry in gauge theories at finite temperatures is discussed. This symmetry is studied in terms of A_0-effective potential. We consider two- and four-dimensional models where the question on physical interpretation of minima of $A_0$-effective potential can be considered exactly. It is shown that the correct result for the partition function can be obtained by summation of contributions near all minima of the $A_0$-effective potential.