Quasihyperbolic spaces and Frobenius groups of finite Morley rank (joint work with Katrin Tent)
A Frobenius group is a group G together with a proper nontrivial malnormal subgroup H. A classical result due to Frobenius states that finite Frobenius groups split, i.e. they can be written as a semidirect product of a normal subgroup and the subgroup H. It is an open question if this holds true for groups of finite Morley rank, and the existence of a non-split Frobenius group of finite Morley rank would contradict the Algebraicity Conjecture. Only partial results are known. By a recent result due to Frecon, so called bad groups cannot exist in Morley rank 3. Frecon’s proof utilizes a point-line geometry defined on the group. A geometry with similar properties but defined on the set of involutions can be used to study sharply 2-transitive groups of Morley rank 6. By axiomatizing these geometries, we are able to extend the above results to other classes of Frobenius groups of finite Morley rank and to provide new criteria for splitting.
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