  Symmetry protected topological (SPT) phases are gapped quantum phases with a
certain symmetry, which can all be smoothly connected to the same trivial
product state if we break the symmetry. For non-interacting fermion systems
with time reversal (T), charge conjugation (C), and/or U(1) (N) symmetries, the
total symmetry group can depend on the relations between those symmetry
operations, such as T N T^{-1}= N or T N T^{-1}= -N. As a result, the SPT
phases of those fermion systems with different symmetry groups have different
classifications. In this paper, we use Kitaev's K-theory approach to classify
the gapped free fermion phases for those possible symmetry groups. In
particular, we can view the U(1) as a spin rotation. We find that
superconductors with the S_z spin rotation symmetry are classified by Z in even
dimensions, while superconductors with the time reversal plus the S_z spin
rotation symmetries are classified by Z in odd dimensions. We show that all 10
classes of gapped free fermion phases can be realized by electron systems with
certain symmetries. We also point out that to properly describe the symmetry of
a fermionic system, we need to specify its full symmetry group that includes
the fermion number parity transformation (-)^N. The full symmetry group is
actually a projective symmetry group.
