We present analysis of the evolution of dark matter halos in dense environm
ents of groups and clusters in dissipationless cosmological simulations. Th
e premature destruction of halos in such environments, known as "the overme
rging," reduces the predictive power of N-body simulations and makes diffic
ult any comparison between models and observations. We analyze the possible
processes that cause the overmerging and assess the extent to which this p
roblem can be cured with current computer resources and codes. Using both a
nalytic estimates and high-resolution numerical simulations, we argue that
the overmerging is mainly due to the lack of numerical resolution. We find
that the force and mass resolution required for a simulated halo to survive
in galaxy groups and clusters is extremely high and was almost never reach
ed before: similar to 1-3 kpc and 10(8)-10(9) M., respectively. We use the
high-resolution Adaptive Refinement Tree (ART) N-body code to run cosmologi
cal simulations with particle mass approximate to 2 x 10(8) h(-1) M. and sp
atial resolution approximate to 1-2 h(-1) kpc and show that in these simula
tions the halos do survive in regions that would appear overmerged with low
er force resolution. Nevertheless, the halo identification in very dense en
vironments remains a challenge even with resolution this high. We present t
wo new halo-finding algorithms developed to identify both isolated and sate
llite halos that are stable (existed at previous moments) and gravitational
ly bound. To illustrate the use of the satellite halos that survive the ove
rmerging, we present a series of halo statistics, which can be compared wit
h those of observed galaxies. Particularly, we find that, on average, halos
in groups have the same velocity dispersion as the dark matter particles;
i.e., they do not exhibit significant velocity bias. The small-scale (100 k
pc to 1 Mpc) halo correlation function in both models is well described by
the power law xi proportional to r(-1.7) and is in good agreement with obse
rvations. It is slightly antibiased (b approximate to 0.7-0.9) relative to
the dark matter. To test other galaxy statistics, we use the maximum of the
halo rotation velocity and the Tully-Fisher relation to assign luminosity
to the halos. For two cosmological models, a flat model with the cosmologic
al constant and Omega(0) = 1 - Omega(Lambda) = 0.3, h = 0.7 and a model wit
h a mixture of cold and hot dark matter and Omega(0) = 1.0, Omega(v) = 0.2,
h = 0.5, we construct luminosity functions and evaluate mass-to-light rati
os in groups. Both models produce luminosity functions and mass-to-light ra
tios (similar to 200-400) that are in reasonable agreement with observation
s. The latter implies that the mass-to-light ratio in galaxy groups (at lea
st for M-vir less than or similar to 3 x 10(13) h(-1) M. analyzed here) is
not a good indicator of Omega(0).