  We describe the properties of a sample of galaxy groups with very unusual
distributions of galaxy luminosities. The most extreme example has an X-ray
luminosity similar to that of the Virgo cluster but has a very low richness,
with only one galaxy brighter than L*, compared with six in Virgo. That one
galaxy, however, is optically more luminous than any galaxy in Virgo and has an
optical luminosity as bright as many of the central cD galaxies in rich Abell
clusters. The characteristic feature of the fossil groups we study is that most
of the light arises from one dominant, central galaxy. We define a fossil
system and, based on this definition, construct a small X-ray selected,
flux-limited sample of fossil groups with well known selection criteria. We
confirm that these systems are indeed groups of galaxies, but dominated by one
central luminous giant elliptical galaxy and with few, or no, L* galaxies. We
find that fossil systems represent 8%-20% of all systems of the same X-ray
luminosity. Fossil groups are at least as numerous as all poor and rich
clusters combined, and are thus a possible site for the formation of luminous
central cluster galaxies before infall into clusters occurs. The fossil systems
in our sample have significantly higher X-ray luminosities than normal groups
of similar total optical luminosities (or similar X-ray temperature, where the
latter can be measured). These enhanced X-ray luminosities may be due to
relatively cool gas in the innermost regions or due to a low central gas
entropy. We interpret fossil groups as old, undisturbed systems which have
avoided infall into clusters, but where galaxy merging of most of the L*
galaxies has occurred. An early formation epoch, before that of most groups,
could explain low central gas entropies and high X-ray luminosities.
