  We consider the influence of an external periodic potential on the fractional
quantum Hall effect of two-dimensional interacting electron systems. For many
electrons on a torus, we find that the splitting of incompressible ground state
degeneracies by a weak external potential diminishes as $\exp ( - L/ \xi)$ at
large system size $L$. We present numerical results consistent with a scenario
in which $\xi$ diverges at continuous phase transitions from fractional to
integer quantum Hall states which occur with increasing external potential
strength.
