A small survey of the UV-absorbing gas in 12 low-z galaxy groups has been conducted using the Cosmic Origins Spectrograph on board the Hubble Space Telescope. Targets were selected from a large, homogeneously selected sample of groups found in the Sloan Digital Sky Survey. A critical selection criterion excluded sight lines that pass close (<1.5 virial radii) to a group galaxy, to ensure absorber association with the group as a whole. Deeper galaxy redshift observations are used both to search for closer galaxies and also to characterize these 10^13.5^-10^14.5^M_{sun}_ groups, the most massive of which are highly virialized with numerous early-type galaxies (ETGs). This sample also includes two spiral-rich groups, not yet fully virialized. At group-centric impact parameters of 0.3-2Mpc, these signal-to-noise ratios = 15-30 spectra detected HI absorption in 7 of 12 groups; high (OVI) and low (SiIII) ion metal lines are present in two-thirds of the absorption components. None of the three most highly virialized, ETG-dominated groups are detected in absorption. Covering fractions >~50% are seen at all impact parameters probed, but do not require large filling factors despite an enormous extent. Unlike halo clouds in individual galaxies, group absorbers have radial velocities that are too low to escape the group potential well without doubt. This suggests that these groups are "closed boxes" for galactic evolution in the current epoch. Evidence is presented that the cool and warm group absorbers are not a pervasive intra-group medium (IGrM), requiring a hotter (T~10^6^-10^7^K) IGrM to be present to close the baryon accounting.