Groups of finite Morley rank with a pseudoreflection action

Abstract

In this work, we give two characterisations of the general linear group as a group G of finite Morley rank acting on an abelian connected group V of finite Morley rank definably, faithfully and irreducibly. To be more precise, we prove that if the pseudoreflection rank of G is equal to the Morley rank of V , then V has a vector space structure over an algebraically closed field, G ∼ = GL(V ) and the action is the natural action. The same result holds also under the assumption of Prüfer 2-rank of G being equal to the Morley rank of V .

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

“…The following three results from our earlier paper [5] will be useful in this work as well. Lemma 7.1 in [5] was stated under stronger assumptions on V; however, the proof used only the fact that V is connected, abelian and has no involutions. So we state Lemma 7.1 in this stronger form: Moreover, (b) for each V i , i = 1, .…”

“…Groups of finite Morley rank equipped with a definable action arise naturally as binding groups in many first order theories: for example, Lie groups of the Picard-Vessiot theory of linear differential equations can be viewed as a special case [40]. A more detailed discussion of binding groups that play in model theory a role akin to that of Galois groups could be found in the Introduction to [5].…”

“…The proof of Theorem 1 appears to be deceptively self-contained; indeed it uses only the general theory of groups of finite Morley rank, with two notable exceptions: the basis of induction, the case n 3, is a combination of delicate work by Deloro [27] and a difficult result by Borovik and Deloro [12] that uses almost all existing machinery of the classification of groups of odd type, while the final identification of the group G with GL(V) uses our previous result [5]; the proof of the latter required revisiting the older stages of the classification of groups of finite Morley rank [4].…”

“…To exclude the case when G in our Theorem 1 is not connected, we will need a result which uses concepts from one of our earlier papers [5]. A special case of this result, when G is connected, was stated as [5,Corollary 1.3].…”

“…Observe first that rk F n = n implies rk F = 1. Pseudoreflection subgroups in the sense of [5] are connected definable abelian subgroups R < G ♯ such that V = [V, R] ⊕ C V (R) and R acts transitively on the non-zero elements of [V, R]. By Fact 2.9, one can immediately conclude that R ≃ F * and [V, R] ≃ F + .…”