The ground state properties of a single-component one-dimensional Coulomb gas are investigated. We use Bose-Fermi mapping for the ground state wave function which permits to solve the Fermi sign problem in the following respects (i) the nodal surface is known, permitting exact calculations (ii) evaluation of determinants is avoided, reducing the numerical complexity to that of a bosonic system, thus allowing simulation of a large number of fermions. Due to the mapping the energy and local properties in one-dimensional Coulomb systems are exactly the same for Bose-Einstein and Fermi-Dirac statistics. The exact ground state energy has been calculated in homogeneous and trapped geometries by using the diffusion Monte Carlo method. We show that in the low-density Wigner crystal limit an elementary low-lying excitation is a plasmon, which is to be contrasted with the large-density ideal Fermi gas/Tonks-Girardeau limit, where low lying excitations are phonons. Exact density profiles are confronted to the ones calculated within the local density approximation which predicts a change from a semicircular to inverted parabolic shape of the density profile as the value of the charge is increased.
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