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

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 ⩾ n . We conclude that m = n , and the action is equivalent to the natural action of GL 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.