  An extra Abelian gauge symmetry is motivated in many new physics models in
both supersymmetric and nonsupersymmetric cases. Such a new gauge symmetry may
interact with both the observable sector and the hidden sector. We
systematically investigate the most general residual discrete symmetries in
both sectors from a common Abelian gauge symmetry. Those discrete symmetries
can ensure the stability of the proton and the dark matter candidate. A hidden
sector dark matter candidate (lightest U-parity particle or LUP) interacts with
the standard model fields through the gauge boson Z', which may selectively
couple to quarks or leptons only. We make a comment on the implications of the
discrete symmetry and the leptonically coupling dark matter candidate, which
has been highlighted recently due to the possibility of the simultaneous
explanation of the DAMA and the PAMELA results. We also show how to construct
the most general U(1) charges for a given discrete symmetry, and discuss the
relation between the U(1) gauge symmetry and R-parity.
