order of the phase transition were determined explicitly. We also pointed out that the phase structure could provide a new approach to the gauge hierarchy problem in grand unified theory (GUT).

This paper is a generalization of the previous work. In particular, we shall investigate the phase structure of an $SU(N)$ gauge-Higgs model with $N_F$ massless fermions on $M^3\otimes S^1$. In the next section we analyze the phase structure of the model, in which both the fermion and Higgs fields belong to the fundamental representation under $SU(N)$. We will find that the phase structure of the model is very different, depending on whether $N$ is even or odd. For the case $N=$ odd, there are the three phases, and the structure is similar to the one obtained in the previous paper. Only two phases, the Hosotani and the coexisting phases, appear for $N=$ even, and the Higgs phase does not exist for finite sizes of the extra dimension. We also determine the critical radius and the order of the phase transition. In the models, the Hosotani mechanism works as the restoration of the gauge symmetry. In Sec. $3$, we consider the case that the representation of matter fields under the 