  A two-dimensional quantum Hall system without disorder for a wide class of
interactions including any two-body interaction with finite range is studied by
using the Lieb-Schultz-Mattis method [{\it Ann. Phys. (N.Y.)} {\bf 16}: 407
(1961)]. The model is defined on an infinitely long strip with a fixed large
width, and the Hilbert space is restricted to the lowest $(n_{\rm max}+1)$
Landau levels with a large integer $n_{\rm max}$. We proved that, for a
non-integer filling $\nu$ of the Landau levels, either (i) there is a symmetry
breaking at zero temperature or (ii) there is only one infinite-volume ground
state with a gapless excitation. We also proved the following two theorems: (a)
If a pure infinite-volume ground state has a non-zero excitation gap for a
non-integer filling $\nu$, then a translational symmetry breaking occurs at
zero temperature. (b) Suppose that there is no non-translationally invariant
infinite-volume ground state. Then, if a pure infinite-volume ground state has
a non-zero excitation gap, the filling factor $\nu$ must be equal to a rational
number. Here the ground state is allowed to have a periodic structure which is
a consequence of the translational symmetry breaking. We also discuss the
relation between our results and the quantized Hall conductance, and
phenomenologically explain why odd denominators of filling fractions $\nu$
giving the quantized Hall conductance, are favored exclusively.
