  We investigate the use of the halo mass-gap statistic --- defined as the
logarithmic difference in mass between the host halo and its most massive
satellite subhalo --- as a probe of halo age and concentration. A cosmological
N-body simulation is used to study N ~25, 000 group/cluster sized halos in the
mass range 10^12.5 < M_halo/M_sol < 10^14.5. In agreement with previous work,
we find that halo mass-gap is related to halo formation time and concentration.
On average, older and more highly concentrated halos have larger halo
mass-gaps, and this trend is stronger than the mass-concentration relation over
a similar dynamic range. However, there is a large amount of scatter owing to
the transitory nature of the satellite subhalo population, which limits the use
of the halo mass-gap statistic on an object-by-object basis. For example, we
find that 20% of very large halo mass-gap systems (akin to "fossil groups") are
young, and have likely experienced a recent merger between a massive satellite
subhalo and the central subhalo. We relate halo mass-gap to the observable
stellar mass-gap via abundance matching. Using a galaxy group catalog
constructed from the Sloan Digital Sky Survey Data Release 7, we find that the
star formation and structural properties of galaxies at fixed mass show no
trend with stellar mass-gap. This is despite a variation in halo-age of ~2.5
Gyr over ~1.2 dex in stellar mass-gap. Thus, we find no evidence to suggest
that the halo formation history significantly affects galaxy properties.
