A central decomposition of connected groups of finite Morley rank

Abstract We introduce a simple equivalence relation on strongly minimal sets in a structure of finite Morley rank, which corresponds, in stability theory, to the non-orthogonality of the associated types. We use it in a group 𝐺 of finite Morley rank to define, for each strongly minimal set 𝑋, two connected normal subgroups <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>M</m:mi> <m:mi>G</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>X</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> M_{G}(X) and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>W</m:mi> <m:mi>G</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>X</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> W_{G}(X) . When 𝐺 is connected, these subgroups provide a central decomposition of 𝐺 that yields a direct product decomposition of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:mi>Z</m:mi> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> G/Z(G) into unidimensional factors, as well as a central decomposition of its derived subgroup into unidimensional subgroups.