  We study several different $Z_2$ topological ordered states in frustrated
spin systems. The effective theories for those different Z_2 topological orders
all have the same form -- a Z_2 gauge theory which can also be written as a
mutual U(1) x U(1) Chern-Simons theory. However, we find that the different Z_2
topological orders are reflected in different projective realizations of
lattice symmetry in the same effective mutual Chern-Simons theory. This result
is obtained by comparing the ground-state degeneracy, the ground-state quantum
numbers, the gapless edge state, and the projective symmetry group of
quasi-particles calculated from the slave-particle theory and from the
effective mutual Chern-Simons theories. Our study reveals intricate relations
between topological order and symmetry.
