Investigating the energy region between 1017 eV and 4 × 1018 eV for primary cosmic particles will lead to a deeper understanding of the origin of cosmic rays. Effects of the transition from galactic to extragalactic origin are expected to be visible in this region. The knowledge of the composition of cosmic rays strongly depends on the hadronic interaction models, which are applied in the air shower reconstruction. Directly determining the number of muons from an air shower on ground level will improve the precision of the composition measurements by reducing the dependence on the models.The Pierre Auger Observatory is facing these challenges with an upgrade of the original detector setup. A denser sub-array of water Cherenkov detectors and a dedicated muon detector (MD) array constitute the AMIGA enhancement (Auger Muon and Infill for the Ground Array). Additional fluorescence telescopes constitute HEAT (High Elevation Auger Telescopes).Seven MD modules have been installed until mid 2012 in a first hexagon at the site of the Pierre Auger Observatory in Malargüe, Argentina. The corresponding readout electronics, and 19 more of these setups, were assembled and tested in Siegen to assure correct functionality. The detectors were incorporated in the trigger structure of the original surface detector (SD) array of the Pierre Auger Observatory and are now taking data synchronously.In the framework of this thesis, system tests have been developed, a pre-unitary cell (PUC) of seven modules has been successfully operated and their trigger has been synchronised with the SD trigger. First data from the MD have been analysed and have been combined with data from the SD.