  We identify galaxy groups and clusters in volume-limited samples of the SDSS
redshift survey, using a redshift-space friends-of-friends algorithm. We
optimize the friends-of-friends linking lengths to recover galaxy systems that
occupy the same dark matter halos, using a set of mock catalogs created by
populating halos of N-body simulations with galaxies. Extensive tests with
these mock catalogs show that no combination of perpendicular and line-of-sight
linking lengths is able to yield groups and clusters that simultaneously
recover the true halo multiplicity function, projected size distribution, and
velocity dispersion. We adopt a linking length combination that yields, for
galaxy groups with ten or more members: a group multiplicity function that is
unbiased with respect to the true halo multiplicity function; an unbiased
median relation between the multiplicities of groups and their associated
halos; a spurious group fraction of less than ~1%; a halo completeness of more
than ~97%; the correct projected size distribution as a function of
multiplicity; and a velocity dispersion distribution that is ~20% too low at
all multiplicities. These results hold over a range of mock catalogs that use
different input recipes of populating halos with galaxies. We apply our
group-finding algorithm to the SDSS data and obtain three group and cluster
catalogs for three volume-limited samples that cover 3495.1 square degrees on
the sky. We correct for incompleteness caused by fiber collisions and survey
edges, and obtain measurements of the group multiplicity function, with errors
calculated from realistic mock catalogs. These multiplicity function
measurements provide a key constraint on the relation between galaxy
populations and dark matter halos.
