  (Abridged) We demonstrate that the tenet of hierarchical structure growth
leads directly to a robust, falsifiable prediction for the correlation between
stellar fraction (fstar) and total system mass (M500) of galaxy groups and
clusters. This prediction is relatively insensitive to the details of baryonic
physics or cosmological parameters. In particular, if the fstar-M500 relation
is fixed and does not evolve with redshift, CDM models predict the logarithmic
slope of this relation to be b>-0.3. This constraint can be weakened if the
fstar-M500 relation evolves strongly, but this implies more stars must be
formed in situ in groups at low redshift. Conservatively requiring that at
least half the stars in groups were formed by z=1, the constraint from
evolution models is b>-0.35. Since the most massive clusters (M500=1E15 Msun)
are observed to have fstar=0.01, this means that groups with M500=5E13 Msun
must have fstar<0.03. Recent observations by Gonzalez et al. (2007) indicate a
much steeper relation, with fstar>0.04 in groups, leading to b=-0.64. If
confirmed, this would rule out hierarchical structure formation models: today's
clusters could not have been built from today's groups, or even from the
higher-redshift progenitors of those groups. We perform a careful analysis of
these and other data to identify the most important systematic uncertainties in
their measurements. Although correlated uncertainties on stellar and total
masses might explain the steep observed relation, the data are only consistent
with theory if the observed group masses are systematically underestimated.
