Groups of Mixed Type

The Classical Involution Theorem for Groups of Finite Morley Rank

“…w x The first division appeared in a work of Altınel et al 2 . Later Jaligot proved a version of this result which does not use the tameness assumption.…”

“…Long root SL -subgroups are always isomorphic to SL , 2 2 while short root SL -subgroups in adjoint type B -groups are isomorphic 2 n to PSL . 2 An algebraic group is generated by its root SL -subgroups. In a simple 2 algebraic group, all long root SL -subgroups are conjugate to each other, 2 and similarly all short root SL -subgroups are conjugate to each other.…”

“…2 An algebraic group is generated by its root SL -subgroups. In a simple 2 algebraic group, all long root SL -subgroups are conjugate to each other, 2 and similarly all short root SL -subgroups are conjugate to each other. 2 By the description of the centralizers of involutions in a simple algebraic Ž w x w x. group G of characteristic different from 2 see 22 and compare with 8 that is not PSL , it easily follows that the long root SL -subgroups are 2 2 classical subgroups of G. Conversely, with the exception of G , a classical 2 subgroup in G is conjugate to a long root subgroup.…”

“…Hence G has two conjugacy classes of classical subgroups. 2 The most important fact for our purposes is the following property of classical subgroups in simple algebraic groups; this property immediately follows from the characterization of these groups as long root SL -sub-2 groups. Note that if G is a simple algebraic group, z is a classical Ž .…”

“…In the group G , the centralizer of an arbitrary involution is isomorphic 2 to SL ) SL , central product with amalgamated centers, where one factor 2 2 corresponds to a long root SL -subgroup and the other to a short root 2 Ž . SL -subgroup so the factors are not conjugate .…”