the fermions in the adjoint representation, the residual gauge symmetry is $U(1)^{N-1}$, irrespective of the flat directions.

Conclusions and Discussions
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We have study the phase structure of the $SU(N)$ gauge-Higgs models with $N_F$ massless fermions on the space-time $M^3\otimes S^1$. There are two kinds of the order parameters for gauge symmetry breaking in the models. One is the vacuum expectation value of the Higgs field $\vev{\Phi}$ (the Higgs mechanism) and the other is the vacuum expectation value of the component gauge field for the $S^1$ direction $\vev{A_y}$ (the Hosotani mechanism). The former works at the tree level, while the latter is effective at the quantum level and sensitive to the size of $S^1$. There is also the interaction between $\vev{A_y}$ and $\vev{\Phi}$, which depends on the size as well. Thus, the dominant contribution to the effective potential comes from the different physical origins, depending on the size of the extra dimension. Therefore, the phase structure depends on the size (in addition to the parameters of the models) in general. This is expected to be a general feature in gauge-Higgs models on the space-time.

We have computed the effective potential for the two kinds of the order parameters in a 