Simple \(L^*\)-groups of even type with strongly embedded subgroups.
The paper is a contribution to the classification of infinite simple groups of finite Morley rank. The main open conjecture here, the Cherlin-Zilber algebraicity conjecture, is that every such group is algebraic over an algebraically closed field. A group of finite Morley rank is said to be (1) of even type if its Sylow 2-subgroups are infinite of bounded exponent; (2) of odd type if its Sylow 2-subgroups are finite extensions of nontrivial 2-tori; (3) of degenerate type if its Sylow 2-subgroups are finite. A group of finite Morley rank is called an \(L^*\)-group if every of its proper infinite definable simple sections is either an algebraic group over an algebraically closed field, or of odd or degenerate type. The verification of the algebraicity conjecture for the simple \(L^*\)-groups of even type would prove the algebraicity conjecture for all simple groups of finite Morley rank of even type. The authors prove the following partial result. Let \(G\) be a simple \(L^*\)-group of finite Morley rank of even type. Assume that \(G\) has a strongly embedded subgroup \(M\) such that \(C_G(A,B)\) is finite whenever \(A\) and \(B\) are two distinct conjugates of \(\Omega_1(M)\). Then \(G\simeq\text{PSL}_2(F)\), where \(F\) is an algebraically closed field of characteristic~2. (Here \(M\) is said to be strongly embedded if \(M\) is a proper definable subgroup such that \(M\) contains involutions but for any \(g\in G\setminus M\) the intersection \(M\cap M^g\) has no involutions; and \(\Omega_1(M)\) denotes the subgroup generated by all involutions in \(M\).)
scientific article; zbMATH DE number 2057666
scientific article