Structure of linear groups definable in a field of finite Morley rank.
The author uses \textit{B. Poizat}'s identification of simple linear groups of finite Morley rank in positive characteristic as algebraic groups [J. Symb. Log. 66, No. 4, 1637-1646 (2001; Zbl 1005.03039)] and an analysis of unipotent groups in characteristic zero to prove a structure theorem for a linear group \(G\) of finite Morley rank: In positive characteristic \(G/R(G)\) is isomorphic to a finite product of simple algebraic groups, in characteristic zero \(G/R(G)\) is the product of a definable group without unipotent elements and finitely many simple algebraic groups. As a consequence he obtains conjugacy of the Borel and Carter subgroups, as well as of the maximal tori. Similar results hold for linear groups of Morley rank \(\omega n\) and characteristic zero, provided no groups of finite rank are definable.
scientific article; zbMATH DE number 2119121