  We construct mock catalogs of galaxy groups using subhalo abundance matching
(SHAM) and undertake several new tests of the SHAM prescription for the
galaxy-dark matter connection. All SHAM models we studied exhibit significant
tension with galaxy groups observed in the Sloan Digital Sky Survey (SDSS). The
SHAM prediction for the field galaxy luminosity function (LF) is systematically
too dim, and the group galaxy LF systematically too bright, regardless of the
details of the SHAM prescription. SHAM models connecting r-band luminosity, Mr,
to Vacc, the maximum circular velocity of a subhalo at the time of accretion
onto the host, faithfully reproduce galaxy group abundance as a function of
richness, g(N). However, SHAM models connecting Mr with Vpeak, the peak value
of Vmax over the entire merger history of the halo, over-predict galaxy group
abundance. Our results suggest that no SHAM model can simultaneously reproduce
the observed g(N) and two-point projected galaxy clustering. Nevertheless, we
also report a new success of SHAM: an accurate prediction for Phi(m12), the
abundance of galaxy groups as a function of magnitude gap m12, defined as the
difference between the r-band absolute magnitude of the two brightest group
members. We show that it may be possible to use joint measurements of g(N) and
Phi(m12) to tightly constrain the details of the SHAM implementation.
Additionally, we show that the hypothesis that the luminosity gap is
constructed via random draws from a universal LF provides a poor description of
the data, contradicting recent claims in the literature. Finally, we test a
common assumption of the Conditional Luminosity Function (CLF) formalism, that
the satellite LF need only be conditioned by the brightness of the central
galaxy. We find this assumption to be well-supported by the observed Phi(m12).
