21.19 (2026)

Suppose that $S$ and $M$ are groups of finite Morley rank, $S$ is an infinite group, and $M$ is a non-trivial connected group definably and faithfully acting on $S$. This action is said to be irreducible if $M$ does not leave invariant any definable non-trivial proper subgroup of $S$. Prove that if $S$ is a simple group such that every proper definable subgroup of $S$ is nilpotent, and the action of $M$ on $S$ is irreducible, then this action is equivalent to the action of $S$ on itself by conjugation.