We present analysis of the evolution of dark matter halos in dense environments of groups and clusters in dissipationless cosmological simulations. The premature destruction of halos in such environments, known as "the overmerging," reduces the predictive power of N-body simulations and makes difficult any comparison between models and observations. We analyze the possible processes that cause the overmerging and assess the extent to which this problem can be cured with current computer resources and codes. Using both analytic 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 reached 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 cosmological simulations with particle mass approximate to 2 x 10(8) h(-1) M. and spatial resolution approximate to 1-2 h(-1) kpc and show that in these simulations the halos do survive in regions that would appear overmerged with lower force resolution. Nevertheless, the halo identification in very dense environments remains a challenge even with resolution this high. We present two new halo-finding algorithms developed to identify both isolated and satellite halos that are stable (existed at previous moments) and gravitationally bound. To illustrate the use of the satellite halos that survive the overmerging, we present a series of halo statistics, which can be compared with 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 kpc 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 observations. 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 cosmological constant and Omega(0) = 1 - Omega(Lambda) = 0.3, h = 0.7 and a model with 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 ratios in groups. Both models produce luminosity functions and mass-to-light ratios (similar to 200-400) that are in reasonable agreement with observations. The latter implies that the mass-to-light ratio in galaxy groups (at least 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).
