Actions of Groups of Finite Morley Rank on Small Abelian Groups

Abstract

We classify actions of groups of finite Morley rank on abelian groups of Morley rank 2: there are essentially two, namely the natural actions of SL(V) and GL(V) with V a vector space of dimension 2. We also prove an identification theorem for the natural module of SL2 in the finite Morley rank category.

Permutation groups of finite Morley rank

“…Fact (Deloro [28]). Let (G, V ) be a faithful, irreducible module of finite Morley rank where G is connected and V has rank 2.…”

The Structure of an SL2‐module of finite Morley rank

“…The challenge of the study of SL 2 -modules of finite Morley rank is to classify irreducible SL 2 -representations of finite Morley rank. This is what both Cherlin and Deloro in [3] and Deloro in [4] have done with some limitations on the Morley rank of V , and we hope this article to be a contribution to such a respectable project.…”

“…Deloro showed that only assuming , the abelian group V is the natural G ‐module , and in particular a vector space over . With Cherlin in , another linear representation is identified when the limitation is pushed to with some additional assumptions of G ‐minimality of V and faithfulness of the action: Fact In a universe of finite Morley rank, consider the following definable objects: a field , a group , an abelian group V , and a faithful action of G on V for which V is G ‐minimal.…”

“…Deloro showed that only assuming rk(V ) ≤ 2rk(K), the abelian group V is the natural G-module [4], and in particular a vector space over K. With Cherlin in [3], another linear representation is identified when the limitation is pushed to rk(V ) ≤ 3rk(K) with some additional assumptions of G-minimality of V and faithfulness of the action:…”

Recognizing PGL$_3$ via generic 4-transitivity

“…If the action is by automorphisms on an abelian group of rank 2, then [Del09] shows that one does not exceed generic 2-transitivity. The main result in the general rank 2 setting is due to Gropp in [Gro92] where the author shows that if one assumes generic sharp n-transitivity then n is at most 5.…”