We study the improper Bogoliubov transformation which is an improper rotation in the SO(2N) group in Fermion systems. The mathematical properties of the Bogoliubov transformation are investigated generally, when the dimension N is odd. It is shown that the improper transformation appears naturally and some quasi-particles become gapless. This leads to the property that some physical quantities have singularities in momentum space across the Fermi surface. Moreover it is pointed out that such a case may be realized in quark matter.