Pseudo-tori and subtame groups of finite Morley rank

Conjugacy of Carter Subgroups in Groups of Finite Morley Rank

“…In [7], G. Cherlin defines decent tori as an analogue to tori algebraic. In [12], we introduce pseudo-tori, as a more general notion, independant of the torsion. Here we relate just the more general results used in this paper, and concerning pseudo-tori, but their proof are very often similar to their analogue in [7] concerning decent tori.…”

“…We introduce the notion of U -groups as a more precise unipotence notion than U -groups defined in [9]. This concept depend on pseudo-tori [12]. Notation 3.7.…”

“…Indeed E. Jaligot proved that generous Carter subgroups are conjugate in any group of finite Morley rank (Fact 1.4), consequently our study will only concern non generous Carter subgroups. Moreover the analysis by G. Cherlin in [7] about decent tori (Definition 2.17) allows us to content our study to Carter subgroups without a nontrivial decent torus, and even without a nontrivial pseudo-torus by the generalisation of this work in [12].…”

“…Actually, this theorem is the outcome of several studies concerning the conjugacy of Carter subgroups in groups of finite Morley rank. Indeed, the conjugacy of Carter subgroups has been proven in several important particular cases (see [10,12,15,19]). Notably two of these ones will be very useful for us.…”

Steinberg's torsion theorem in the context of groups of finite Morley rank

“…Decent tori are definable hulls of abelian, divisible, torsion groups. As their centralizers in a connected group are connected ([1, Theorem 1]; see also [4,Corollary 3.9]), so are those of p-tori.…”