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We investigate the configuration where a group of finite Morley rank actsdefinably and generically $m$-transitively on an elementary abelian $p$-groupof Morley rank $n$, where $p$ is an odd prime, and $m\geqslant n$. We concludethat $m=n$, and the action is equivalent to the natural action of$\operatorname{GL}_n(F)$ on $F^n$ for some algebraically closed field $F$. Thisstrengthens our earlier result in arXiv:1802.05222, and partially answers twoproblems posed in [9].

Groups of finite Morley rank with a generically multiply transitive action on an abelian group