  We investigate by means of Monte Carlo simulations the fully connected
p-state Potts model for different system sizes in order to see how the static
and dynamic properties of a finite model compare with the, exactly known,
behavior of the system in the thermodynamic limit. Using p=10 we are able to
study the equilibrium dynamics for system sizes as large as N=2560. We find
that the static quantities, such as the energy, the entropy, the spin glass
susceptibility as well as the distribution of the order parameter P(q) show
very strong finite size effects. From P(q) we calculate the forth order
cumulant g_4(N,T) and the Guerra parameter G(N,T) and show that these
quantities cannot be used to locate the static transition temperature for the
system sizes investigated. Also the spin-autocorrelation function C(t) shows
strong finite size effects in that it does not show a plateau even for
temperatures around the dynamical critical temperature T_D. We show that the
N-and T-dependence of the \alpha-relaxation time can be understood by means of
a dynamical finite size scaling Ansatz. C(t) does not obey the time-temperature
superposition principle for temperatures around T_D, but does so for
significantly lower T. Finally we study the relaxation dynamics of the
individual spins and show that their dependence on time depends strongly on the
chosen spin, i.e. that the system is dynamically very heterogeneous, which
explains the non-exponentiality of C(t).
