Groups of finite Morley rank with a generically sharply multiply transitive action

Abstract

We prove that if G is a group of finite Morley rank that acts definably and generically sharply n-transitively on a connected abelian group V of Morley rank n with no involutions, then there is an algebraically closed field F of characteristic 2 such that V has the structure of a vector space of dimension n over F and G acts on V as the group GL n (F) in its natural action on F n .

Permutation groups of finite Morley rank

“…Conjecture 6 is also confirmed in the important special case when the action of H on V is sharply generically transitive [9]. There is hope that the general case could be reduced to this special result if one obtains good bounds for the Morley ranks of irreducible modules for some specific finite groups, see Problem 11 and its discussion in § 4.3.1.…”

Finite group actions on abelian groups of finite Morley rank

“…My special thanks go to Ayse Berkman, my research collaborator and co-author of many years. Problems solved in this paper originate in our joint project [2,3] which we developed mostly during our visits to the Nesin Mathematics Village in Turkey. 1 We were planning to meet there again during the Easter break and in the summer.…”

“…They are linked to a range of problems on groups of finite Morley rank discussed in [5]. Crucially, these results are needed for the forthcoming work by Ayşe Berkman and myself [3] where we remove the 'sharpness' assumption from [2].…”