  We use a modified version of the halo-based group finder developed by Yang et
al. to select galaxy groups from the Sloan Digital Sky Survey (SDSS DR4). In
the first step, a combination of two methods is used to identify the centers of
potential groups and to estimate their characteristic luminosity. Using an
iterative approach, the adaptive group finder then uses the average
mass-to-light ratios of groups, obtained from the previous iteration, to assign
a tentative mass to each group. This mass is then used to estimate the size and
velocity dispersion of the underlying halo that hosts the group, which in turn
is used to determine group membership in redshift space. Finally, each
individual group is assigned two different halo masses: one based on its
characteristic luminosity, and the other based on its characteristic stellar
mass. Applying the group finder to the SDSS DR4, we obtain 301237 groups in a
broad dynamic range, including systems of isolated galaxies. We use detailed
mock galaxy catalogues constructed for the SDSS DR4 to test the performance of
our group finder in terms of completeness of true members, contamination by
interlopers, and accuracy of the assigned masses. This paper is the first in a
series and focuses on the selection procedure, tests of the reliability of the
group finder, and the basic properties of the group catalogue (e.g. the
mass-to-light ratios, the halo mass to stellar mass ratios, etc.). The group
catalogues including the membership of the groups are available at
http://gax.shao.ac.cn/data/Group.html and
http://www.astro.umass.edu/~xhyang/Group.html
