# A conjecture on Zassenhaus groups

In a 1995 paper by Ali Nesin, "Permutation groups of finite Morley rank", the following conjecture is mentioned:

Conjecture 5 An infinite Zassenhaus group $G$ of finite Morley rank is isomorphic to $\operatorname{PSL}_2(K)$ for some algebraically closed field $K$. Furthermore the action of $G$ is isomorphic to the action of $\operatorname{PSL}_2(K)$ acting naturally on the projective line.

Searching the web for any advances in this conjecture were fruitless, and I even couldn't find another mentioning of this conjecture. My question is whether it is still open, and if so - what are the current, if any, settings in which a variation of this is known. In particular I'm interested in the variation where $G\leq\operatorname{GL}_n(F)$ is a Zassenhaus group, or even a more restrictive setting, where $G\leq\operatorname{GL}_n(F)$ is a sharply $3$-transitive group.
I know that a variation of this result was proved by Tits in 1941, when he showed that a a locally compact connected sharply 3-transitive group is isomorphic to $\operatorname{PSL}_2(\mathbb{R})$ or $\operatorname{PSL}_2(\mathbb{C})$ (there where no assumptions on the Morley rank, so the field could be non algebraically closed)

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I have briefly touched upon this conjecture in my paper "Special abelian Moufang sets of finite Morley rank" with K. Tent (dx.doi.org/10.1515/JGT.2008.040), see Proposition 2.2, where we prove the conjecture for so-called split special Zassenhaus groups. (The notion of being special comes from Timmesfeld's theory of abstract rank one groups.) But I don't think this qualifies as a serious "advance" so I didn't want to post this as an answer ;) –  Tom De Medts Nov 19 '12 at 12:00
Let me also add that the study of Zassenhaus groups is nowadays often tackled using the theory of Moufang sets, so Googling for "Zassenhaus Moufang sets" gives some information about the general problem (but often in a context which has nothing to do with finite Morley rank). –  Tom De Medts Nov 19 '12 at 12:02
@Tom De Medts: Thank you, I will look into it further. The context of finite Morley rank is not important for the things I want to do. I was first introduced to the theory of Moufang sets by Yoav Segev, who referred me to your paper with him. Unfortunately, though, he didn't know of any breakthroughs in this area done lately and he was not sure that there are now new results either. It seems to me that an interesting special case of this conjecture is the case of sharply $3$-transitive groups, but working with Zassenhaus Moufang sets will require additional assumptions on the group. –  Dennis Gulko Nov 19 '12 at 13:01