259 research outputs found

    Simple groups of Morley rank 5 are bad

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    By exploiting the geometry of involutions in NN_\circ^\circ-groups of finite Morley rank, we show that any simple group of Morley rank 55 is a bad group all of whose proper definable connected subgroups are nilpotent of rank at most 22. The main result is then used to catalog the nonsoluble connected groups of Morley rank 55

    A signalizer functor theorem for groups of finite Morley rank

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    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

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    We show that for a wide class of groups of finite Morley rank the presence of a split BNBN-pair of Tits rank 11 forces the group to be of the form PSL2\operatorname{PSL}_2 and the BNBN-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 22-subgroups and show that all such Moufang sets are standard (and thus associated to PSL2(F)\operatorname{PSL}_2(F) for FF an algebraically closed field of characteristic not 22) provided the Hua subgroups are nilpotent. Further, we prove that the same conclusion can be reached whenever the Hua subgroups are LL-groups and the root groups are not simple

    Minimal connected simple groups of finite Morley rank with strongly embedded subgroups

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    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

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    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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