Pushing up and \(C(G,T)\) in groups of finite Morley rank of even type
The long standing Cherlin-Zil'ber Conjecture is that any infinite simple group of finite Morley rank is an algebraic group over an algebraically closed field. The paper under review is a part of a series aimed ultimately at proving the Even Type Conjecture -- a special case of the Cherlin-Zil'ber Conjecture for groups of even type without infinite definable simple sections of degenerate type. Here a group of finite Morley rank is said to be of degenerate (even) type if its Sylow 2-subgroups are finite (infinite of bounded exponent). The authors study simple \(K^*\)-groups of even type. (A group of finite Morley rank is called a \(K^*\)-group if every infinite proper definable simple section of it is algebraic over an algebraically closed field; \(K^*\)-groups naturally arise when one considers minimal counterexamples.) The results are too technical to be formulated here; they are analogs of some results from finite group theory in the context of groups of finite Morley rank (in particular, see \textit{B.~Baumann} [Proc. Am. Math. Soc. 74, 215-222 (1979; Zbl 0409.20009)] and \textit{B.~Stellmacher} [Arch. Math. 46, 8-17 (1985; Zbl 0588.20013)]). In fact, a third of the paper consists of an appendix where a proof of an analog of Stellmacher's result is presented.
scientific article; zbMATH DE number 1740516
scientific article