Abstract

We investigate the configuration where a group of finite Morley rank acts definably and generically $m$-transitively on an elementary abelian $p$-group of Morley rank $n$, where $p$ is an odd prime, and $m\ge n$. We conclude that $m=n$, and the action is equivalent to the natural action of ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_n(F)$ on $F^n$ for some algebraically closed field $F$. This strengthens one of our earlier results, and partially answers two problems posed by Borovik and Cherlin in 2008.

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