NOVELTY - The system (1) has a compression device (9) i.e. exhaust gas turbocharger, for compressing cathode gas. A compressor (10) e.g. radial compressor, is arranged in a cathode gas guide (7). A turbine (12) is connected with the compressor and arranged in a cathode exhaust gas routing system (8). A burner (13) is arranged upstream of the turbine in the routing system. The burner is bridged by a bypass guide (14) such that a degree of bridging is adjusted based on a combustion process. The turbine and the compressor are connected with each other in force-fit manner by a shaft for force transmission.    USE - Hydrogen fuel cell system e.g. stationary fuel cell system and portable fuel cell system, for a motor car.    ADVANTAGE - The turbine and the compressor are connected with each other in force-fit manner by the shaft for the force transmission, thus achieving the simple and inexpensive compression device that requires a limited installation space. The system easily and cost-effectively reduces concentration of pollutants in fuel cell exhaust gas, and improves heat and water management. The system avoids collection of fuel downstream of the burner, thus increasing safety of operation of the system can be enhanced, and avoiding the need for a separate afterburner. The system is inexpensive, and enables precise control of the cathode exhaust gas flowing though the turbine and/or the burner. The system allows simple and inexpensive removal of exhaust gas e.g. nitrogen oxide, and combustion gas e.g. hydrogen, from the exhaust gas without hindering operation of a fuel cell. The system allows precise control of combustion and avoids the need for cost-intensive post treatment of the turbine exhaust gas.    DETAILED DESCRIPTION - An INDEPENDENT CLAIM is also included for a method for operating a fuel cell system.    DESCRIPTION OF DRAWING(S) - The drawing shows a pneumatic connection diagram of a hydrogen fuel cell system.    Hydrogen fuel cell system (1)    Cathode gas guide (7)    Cathode exhaust gas routing system (8)    Compression device (9)    Compressor (10)    Turbine (12)    Burner (13)    Bypass guide (14)