259 research outputs found
Simple groups of Morley rank 5 are bad
By exploiting the geometry of involutions in -groups of finite
Morley rank, we show that any simple group of Morley rank is a bad group
all of whose proper definable connected subgroups are nilpotent of rank at most
. The main result is then used to catalog the nonsoluble connected groups of
Morley rank
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
Moufang sets of finite Morley rank of odd type
We show that for a wide class of groups of finite Morley rank the presence of
a split -pair of Tits rank forces the group to be of the form
and the -pair to be standard. Our approach is via
the theory of Moufang sets. Specifically, we investigate infinite and so-called
hereditarily proper Moufang sets of finite Morley rank in the case where the
little projective group has no infinite elementary abelian -subgroups and
show that all such Moufang sets are standard (and thus associated to
for an algebraically closed field of
characteristic not ) provided the Hua subgroups are nilpotent. Further, we
prove that the same conclusion can be reached whenever the Hua subgroups are
-groups and the root groups are not simple
Minimal connected simple groups of finite Morley rank with strongly embedded subgroups
We show that a minimal nonalgebraic simple groups of finite Morley rank has
Prufer rank at most 2, and eliminates tameness from Cherlin and Jaligot's past
work on minimal simple groups. The argument given here begins with the strongly
embedded minimal simple configuration of Borovik, Burdges and Nesin. The
0-unipotence machinery of Burdges's thesis is used to analyze configurations
involving nonabelian intersections of Borel subgroups. The number theoretic
punchline of Cherlin and Jaligot has been replaced with a new genericity
argument
Model Theory: groups, geometry, and combinatorics
This conference was about recent interactions of model theory with combinatorics, geometric group theory and the theory of valued fields, and the underlying pure model-theoretic developments. Its aim was to report on recent results in the area, and to foster communication between the different communities
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