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We report an inelastic light scattering study of the cyclotron spin-flip mode in the two-dimensional electron system at filling nu=1. The energy of this mode can serve as a probe of the many-body exchange interaction on short length scales. Its magnetic field dependence is compared with predictions based on Hartree-Fock theory. They agree well when including the nonzero width of the electron system. From the measured energies, the exchange enhanced g factor is extracted. It diverges at small fields and differs largely from g factors obtained via transport activation studies.