Involutions in groups of finite Morley rank of degenerate type

$\operatorname{Alt}(n)$ 在无对合的有限 Morley 秩群上的作用

Actions of $\operatorname{Alt}(n)$ on groups of finite Morley rank without involutions

本文研究了有限 Morley 秩群上 $\operatorname{Alt}(n)$ 的忠实作用，证明了无对合时不可解群秩的下界 $n$，并推广到可解群，进而应用于通用传递性限制。

Linear groups of finite Morley rank

We show that a non-algebraic simple group of finite Morley rank with a definable representation over a field has no involutions, and otherwise resembles a bad group. In particular, the modern form of the Cherlin-Zilber alebaricity conjecture hold for such groups.

Simple groups of Morley rank 5 are bad

By exploiting the geometry of involutions in $N_\circ^\circ$-groups of finite Morley rank, we show that any simple group of Morley rank $5$ is a bad group all of whose proper definable connected subgroups are nilpotent of rank at most $2$. The main result is then used to catalog the nonsoluble connected groups of Morley rank $5$.

The geometry of involutions in ranked groups with a TI-subgroup

We revisit the geometry of involutions in groups of finite Morley rank. Our approach unifies and generalises numerous results, both old and recent, that have exploited this geometry; though in fact, we prove much more. We also conjecture that this path leads to a new identification theorem for $\operatorname{PGL}_2(\mathbb{K})$.

On Frattini arguments in L-groups of finite Morley rank

We modify the Frecon-Jaligot construction of Carter subgroups to show that a degenerate type group has a Carter subgroup invariant under the Sylow 2-subgroup of a group of automorphisms; thus reducing the need to know that Carter subgroups are conjugate in degenerate type groups.

Weyl groups of small groups of finite Morley rank

We examine Weyl groups of minimal connected simple groups of finite Morley rank of degenerate type. We show that they are cyclic, and lift isomorphically to subgroups of the ambient group.