Title: The Fitting subgroup of a supersimple group.
Abstract: The Fitting subgroup of a given group G is the subgroup generated by all nilpotent normal subgroups of G. While it is always normal, it may not be nilpotent (neither definable). In fact, the nilpotency of the Fitting group implies its definability. In the model-theoretic paradise of stable groups, Wagner proved that the Fitting subgroup is always nilpotent, generalizing a result of Nesin in the finite Morley rank context. However, this is not known for the wider class of groups with a simple theory.