A Fitting Theorem for Simple Theories

TL;DR

This paper proves that in simple theories, the Fitting subgroup of certain groups is relatively definable and nilpotent, extending to hyperdefinable groups with bounded index and normal nilpotent subgroups.

Contribution

It establishes new structural results about the Fitting subgroup in simple and supersimple theories, including relative definability and nilpotency properties.

Findings

Fitting subgroup is relatively definable and nilpotent in simple theories

In supersimple hyperdefinable groups, the Fitting subgroup has a normal hyperdefinable nilpotent subgroup of bounded index

The Fitting subgroup has bounded index in a hyperdefinable subgroup

Abstract

The Fitting subgroup of a type-definable group in a simple theory is relatively definable and nilpotent. Moreover, the Fitting subgroup of a supersimple hyperdefinable group has a normal hyperdefinable nilpotent subgroup of bounded index, and is itself of bounded index in a hyperdefinable subgroup.