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Background: In atomic nuclei, a concentration of electric dipole strength around the particle threshold, commonly denoted as pygmy dipole resonance, may have a significant impact on nuclear structure properties and astrophysical scenarios. A clear identification of these states and the structure of this resonance is still under discussion. Purpose: We present an experimental and theoretical study of the isospin character of the pygmy dipole resonance and investigation of a splitting of the electric dipole strength previously observed in experiments on N -- 82 nuclei. Method: The pygmy dipole resonance has been studied in the semi-magic Z = 50 nucleus 124Sn by means of the (α, α′;γ) coincidence method at Ea = 136 MeV using the Big-Bite Spectrometer at the Kernfysisch Versneller Instituut in Groningen, The Netherlands. Results: A splitting of the low-energy part of the electric dipole strength was identified in 124Sn by comparing the differential cross sections measured in (α, α′γ) to results stemming from (γ, γ′) photon-scattering experiments. While an energetically lower-lying group of states is observed in both kinds of experiments, a higher-lying group of states is only excited in the (γ, γ′) reaction. In addition, theoretical calculations using the self-consistent relativistic quasiparticle time-blocking approximation and the quasiparticle-phonon model have been performed. Both calculations show a qualitative agreement with the experimental data and predict a low-lying isoscalar component that is dominated by neutron-skin oscillations as expected for the pygmy dipole resonance. Furthermore, the states at higher energies show a pronounced isovector component and a different radial dependence of the corresponding transition densities as expected for the tail of the giant dipole resonance. Conclusions: An experimental signature of the neutron-skin oscillation of the pygmy dipole resonance has been corroborated. The combination of the presented reactions might make it possible to identify states of this resonance. [ABSTRACT FROM AUTHOR]