53 research outputs found
A signalizer functor theorem for groups of finite Morley rank
There is a longstanding conjecture, due to Gregory Cherlin and Boris Zilber,
that all simple groups of finite Morley rank are simple algebraic groups. One
of the major theorems in the area is Borovik's trichotomy theorem. The
"trichotomy" here is a case division of the minimal counterexamples within odd
type, i.e. groups with a divisibble connected component of the Sylow
2-subgroup. We introduce a charateristic zero notion of unipotence which can be
used to obtain a connected nilpotent signalizer functor from any sufficiently
non-trivial solvable signalizer functor. This result plugs seamlessly into
Borovik's work to eliminate the assumption of tameness from his trichotomy
theorem for odd type groups. This work also provides us with a form of
Borovik's theorem for degenerate type groups
Signalizers and balance in groups of finite Morley rank
We show that a minimal counter example to the Cherlin-Zilber Algebraicity
Conjecture for simple groups of finite Morley rank has Prufer 2-rank at most
two. This article covers the signalizer functor theory and identifies the
groups of Lie rank at least three; leaving the uniqueness case analysis to
previous articles. This result signifies the end of the general methods used to
handle large groups; hereafter each individual group PSL, PSL, PSp,
and G will require its own identification theorem
On analogies between algebraic groups and groups of finite Morley rank
We prove that in a connected group of finite Morley rank the centralizers of
decent tori are connected. We then apply this result to the analysis of minimal
connected simple groups of finite Morley rank. Our applications include general
covering properties by Borel subgroups, the description of Weyl groups and the
analysis of toral automorphisms
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