On groups of finite Morley rank of even type
The Cherlin-Zilber conjecture asserts that an uncountably categorical simple group (or, equivalently, a simple group of finite Morley rank) be an algebraic group over an algebraically closed field. Depending on whether its maximal 2-subgroups (2-Sylows) are finite/divisible/of finite exponent/neither of these, such a group is said to be of `degenerate'/`odd'/`even'/`mixed' type; the even type should correspond to an algebraic group over a field of characteristic 2, the odd type to an algebraic group over a field of characteristic \(\not=2\), and the degenerate and mixed types should not exist. An approach to this conjecture initiated by Borovik and Nesin studies a minimal counterexample, and follows the revisionist proof of the classification of finite simple groups. More precisely, a `\(K^*\)-group' is defined to be a group of finite Morley rank whose proper definable simple sections are algebraic groups. Thus a counterexample of minimal Morley rank would be a \(K^*\)-group. However, while \(K^*\)-groups of mixed type are known not to exist and the classification of \(K^*\)-groups of even type is nearing completion, the classification of \(K^*\)-groups of odd type is still incomplete, and \(K^*\)-groups of degenerate type may very possibly exist, thus rendering the induction impossible. The paper under review presents an inductive framework suitable for a resolution of the even type problem, first developed in the first author's habilitation thesis. Define an `\(L^*\)-group' to be a group of finite Morley rank whose proper definable simple sections are either algebraic groups over algebraically closed fields, or of odd or degenerate type. Then all simple groups of finite Morley rank of even type are algebraic if and only if there is no simple non-algebraic \(L^*\)-group of even type. The authors express hope that this can be shown by generalizing the analogous result for \(K^*\)-groups. As a first indication they show Theorem 1. If \(M\) is a weakly embedded subgroup of a simple \(L^*\)-group of even type, then \(M^0/O_2^0(M)\) is of degenerate type. Here \(^0\) denotes the connected component, and \(O_2(M)\) is the maximal normal 2-subgroup. We say that \(M\) is `weakly embedded' if \(M\) has an infinite 2-Sylow, but the 2-Sylows of \(M\cap M^g\) are finite for all \(g\notin M\). Theorem 2. A simple group of finite Morley rank and of even type, whose proper definable simple sections of even type are algebraic, is not of mixed type. Corollary. If all simple groups of finite Morley rank and even type are algebraic, then there is no simple group of finite Morley rank and of mixed type. Analogous results have been at the beginning of the analysis of \(K^*\)-groups of even type, see \textit{T. Altınel} [J. Algebra 180, No. 3, 778-807 (1996; Zbl 0855.20029)], \textit{T. Altınel, A. Borovik, G. Cherlin} [J. Algebra 211, No. 2, 409-456 (1999; Zbl 0921.20037)], and \textit{E. Jaligot} [J. Algebra 212, No. 2, 753-768 (1999; Zbl 0944.20016), ibid. 240, No. 2, 413-444 (2001; Zbl 0994.20028)]; the proofs in the \(L^*\)-case deviate mainly when proving Theorem 1 for `strongly' embedded subgroups (a subgroup \(M\) containing involutions, such that \(M\cap M^g\) has no involution for all \(g\notin M\)). The article has a very useful section on background material, which curiously is placed in the middle.
scientific article; zbMATH DE number 1955952