Most spiral galaxies are found in galaxy groups with low velocity dispersions; most E/S0 galaxies are found in galaxy groups with relatively high velocity dispersions. The mass of the hot gas we can observe in the E/S0 groups via their thermal X-ray emission is, on average, as much as the baryonic mass of the galaxies in these groups. By comparison, galaxy clusters have as much or more hot gas than stellar mass. Hot gas in S-rich groups, however, is of low enough temperature for its X-ray emission to suffer heavy absorption due to Galactic H I and related observational effects and hence is hard to detect. We postulate that such lower temperature hot gas does exist in low velocity dispersion, S-rich groups and explore the consequences of this assumption. For a wide range of metallicity and density, hot gas in S-rich groups can cool in far less than a Hubble time. If such gas exists and can cool, especially when interacting with H I in existing galaxies, then it can help link together a number of disparate observations, both Galactic and extragalactic, that are otherwise difficult to understand.