A two-step fragmentation experiment has been performed at GSI with the RISING setup. It combines the fragment separator FRS, which allows for the production of radioactive heavy ions at relativistic energies, with a high resolution g-spectrometer. This combination offers unique possibilities for nuclear structure investigations like the test of shell model predictions far from stability. Within the present work the question if the N = 14(16)shell stabilisation in Z = 8 oxygen isotopes and the N = 20 shell quenching in 32Mg are symmetric with respect to the isospin projection quantum number Tz has been addressed. New gamma-ray decays were found in the neutron deficient 36Ca and 36K by impinging a radioactive ion beam of 37Ca on a secondary 9Be target. The fragmentation products were selected with the calorimeter telescope CATE and the emitted g-rays were measured with Ge Cluster, MINIBALL, and BaF2 HECTOR detectors. For 36Ca the 2+ -> 0+ transition energy was determined to be 3015(16) keV, which is the heaviest T = 2 nucleus from which gamma-spectroscopic information has been obtained so far. A comparison between the experimental 2+ energies of 36Ca and its mirror nucleus 36S yielded a mirror energy difference of MED = -276(16) keV. In order to understand the large MED value, the experimental single-particle energies from the A=17, T =1/2 mirror nuclei were taken and applied onto modified isospin symmetric USD interactions in shell model calculations. These calculations were in agreement with the experimental result and showed that the experimental single-particle energies may account empirically for the one body part of Thomas-Ehrman and/or Coulomb effects. A method to extract the lifetime of excited states in fragmentation reactions was investigated. Therefore, the dependence between the lifetime of an excited state and the average de-excitation velocity and trajectory of the nuclei in relativistic fragmentation experiments has been studied. Known lifetime values in 34Cl could be confirmed and new values were found for 36K
