  The study of the fractional quantum Hall liquid state of two-dimensional
electrons requires a non-perturbative treatment of interactions. It is possible
to perform exact diagonalizations of the Hamiltonian provided one considers
only a small number of electrons in an appropriate geometry. Many insights have
been obtained in the past from considering electrons moving on a sphere or on a
torus. In the Landau gauge it is also natural to impose periodic boundary
conditions in only one direction, the cylinder geometry. The interacting
problem now looks formally like a one-dimensional problem that can be attacked
by the standard DMRG algorithm. We have studied the efficiency of this
algorithm to study the ground state properties of the electron liquid at lowest
Landau level filling factor $\nu=1/3$ when the interactions are truncated to
the two most important repulsive hard-core components. Use of finite-size DMRG
allows us to conclude that the ground state is a compressible two-electron
bubble phase in agreement with previous Hartree-Fock calculations. We discuss
the treatment of Coulomb interactions in the cylinder geometry. To regularize
the long-distance behavior of the Coulomb potential, we compare two methods:
using a Yukawa potential or forbidding arbitrary long distances by defining the
interelectron distance as the chord distance through the cylinder. This allows
us to observe the Wigner crystal state for small filling factor.
