The formation histories of BCG's and ordinary elliptical galaxies are closely linked. Elliptical galaxies most likely form through the dissipationless merger of smaller spiral or elliptical galaxies (Toomre & Toomre 1972; Toomre 1977),  while the spirals themselves form dissipatively as gas cools radiatively and sinks to the centre of a dark halo (White & Rees 1978). When galaxies collide, dynamical friction combined with strong time-dependent mutual tidal forces redistribute the ordered orbital kinetic energy into random energy allowing the galaxies to merge into an amorphous, triaxial system resembling an elliptical galaxy (Barnes 1988; Hernquist 1992,1993; Barnes & Hernquist 1992ab) . In the various N-body studies of mergers of galaxy pairs, analysis of the merger remnants show that they have light profiles and kinematics similar to observed ellipticals, although the match is not perfect. The simulation of the merger of groups of galaxies (Barnes 1989; Weil & Hernquist 1996) which is more closely related to the formation of BCG's give generally similar results to the merger of galaxy pairs. Simulated merger remnants generally have deVaucouleurs profiles, although the cores can be less dense than real ellipticals when the progenitors are pure stellar disks without a bulge (Hernquist 1992). This problem probably arises from Liouville's theorem and the conservation of fine-grained phase space density (Carlberg 1986). The central phase space density of the remnant can be no greater than that of the progenitors so pure disks with low central phase-space density cannot make ellipticals with a high central value. Disk galaxies with bulges can lead to denser cores (Hernquist 1993), but this only skirts the issue by including an elliptical component with a dense core in the progenitors. The solution to this problem is probably gaseous dissipation through radiative cooling which can lead to higher, central densities in merging galaxies with gas (Kormendy 1989b), although simulations with gas and star formation seem to produce cores which are too dense (Mihos & Hernquist 1994). Despite the uncertainties in core properties, galaxy merging produces remnants with global structure and kinematics similar to real ellipticals and remains the most likely way that they form.
Galaxy clusters hold the key to understanding the formation of elliptical galaxies. While E galaxies only make up about 10% of all galaxies, they are much more abundant in regions of high galaxy density, especially in the centres of clusters of galaxies where they make up most of the galaxy population (Dressler 1980; Dressler 1984). The high frequency of E galaxies in rich clusters has been viewed as a paradox and evidence against the merger hypothesis, since the large relative velocities of galaxies in virialized clusters should not permit galaxy merging (Ostriker 1980). The high number density of galaxies may permit the merger of some galaxies in the low velocity tail of a virialized distribution (Mamon 1992) that may account for the observed elliptical concentration, but this picture does not include the effects of hierarchical collapse. In a cosmological hierarchy, small groups of galaxies will form prior to the collapse and virialization of the cluster and the velocity dispersion in these groups may be low enough to permit dissipationless merging and the formation of elliptical galaxies. Small elliptical galaxies may also be created in gravitational instabilities in the tidal tails of interacting galaxies (Barnes 1992) or from galaxy harassment, the cumulative tidal perturbations from other galaxies or the cluster potential (Moore et al. 1996a). Disk galaxies need a quieter environment to form through dissipative collapse and the centre of a cluster where strong tidal forces from closely interacting protogalaxies is probably the least likely place to form a disk. One might therefore expect elliptical galaxies and in particular a BCG to form in the cluster centre where the lumpy mass flow is converging.
While phenomenological studies of galaxy mergers are useful for establishing generic properties of merger remnants, the initial trajectories are only roughly based on cosmological expectations and the galaxy dark halo models are usually truncated at a smaller mass and extent than seen in halos in cosmological simulations (Dubinski & Carlberg 1991; Navarro, Frenk & White 1996).  An examination of galaxy merging in the cosmological setting of a cluster is therefore necessary to go beyond these studies, though, the large dynamic range in mass between galaxies and clusters (a factor of 1000 or more) make cluster simulation with resolved galaxies difficult to study using either N-body or combined N-body/gasdynamics methods. Galactic dark halos seem to merge too efficiently and form a smooth cluster dark halo with very little internal substructure corresponding to galaxies at late times (Moore et al. 1996b). This problem originates from inadequate dynamic range from large gravitational softening lengths (50 kpc) and too few particles. Larger cluster simulations with particles show that substructure corresponding to a galaxy scale can survive during the formation of a cluster (Carlberg 1994). In N-body gasdynamical simulations, galaxies form in the centre of dark halos through gaseous dissipation. These tightly bound objects may survive infall into the cluster, but in practice, the number of galaxies formed in this way in simulations is highly sensitive to the cooling rate assigned to the gas (Frenk et al. 1996). Gasdynamical simulations are also much more computationally intensive making this method very difficult at present.