Fusion of 2-elements in groups of finite Morley rank (Q2747715)

The Alperin-Goldschmidt fusion theorem [\textit{J. L. Alperin}, J. Algebra 6, 222-241 (1967; Zbl 0168.27202); \textit{D. M. Goldschmidt}, J. Algebra 16, 138-142 (1970; Zbl 0198.04306)] was a useful tool in the classification of the finite simple groups. The theorem says something about conjugation of \(p\)-subgroups in finite groups, but the result is too technical to formulate it in the review. The author claims that similar results are needed in the study of groups of finite Morley rank. He proves some analogs of the theorem for groups of finite Morley rank in the case \(p=2\). The reason of the restriction on \(p\) is that an analog of the Sylow theorem for groups of finite Morley rank is known only for \(p=2\). However, the author shows that for any \(p\) and any group of finite Morley rank the fusion theorem holds if in every definable section of the group the Sylow \(p\)-subgroups are conjugate and locally finite.