We present an optical group catalog between 0.1 < z < 1 based on 16,500 high-quality spectroscopic redshifts in the completed zCOSMOS-bright survey. The catalog published herein contains 1498 groups in total and 192 groups with more than five observed members. The catalog includes both group properties and the identification of the member galaxies. Based on mock catalogs, the completeness and purity of groups with three and more members should be both about 83% with respect to all groups that should have been detectable within the survey, and more than 75% of the groups should exhibit a one-to-one correspondence to the real groups. Particularly at high redshift, there are apparently more galaxies in groups in the COSMOS field than expected from mock catalogs. We detect clear evidence for the growth of cosmic structure over the last seven billion years in the sense that the fraction of galaxies that are found in groups (in volume-limited samples) increases significantly with cosmic time. In the second part of the paper, we develop a method for associating galaxies that only have photo-z to our spectroscopically identified groups. We show that this leads to improved definition of group centers, improved identification of the most massive galaxies in the groups, and improved identification of central and satellite galaxies, where we define the former to be galaxies at the minimum of the gravitational potential wells. Subsamples of centrals and satellites in the groups can be defined with purities up to 80%, while a straight binary classification of all group and non-group galaxies into centrals and satellites achieves purities of 85% and 75%, respectively, for the spectroscopic sample.
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