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The only known finite projective plane with a transitive automorphism group is the Desarguesian plane $PG(2,q)$ and it seems likely that there are no others, although this is not (quite) proved.

However all the papers that I have seen dealing with this problem or variants of this problem say things like "It is a longstanding conjecture that a transitive projective plane is Desarguesian" or "It has been conjectured that ..." and none of them actually say who made the original conjecture.

I've looked in Kantor's papers on flag-transitive planes, Dembowski's book on Finite Geometries, Ostrom and Wagner's paper proving that planes with doubly transitive groups are Desarguesian and Higman and McLaughlin's paper on ABA groups.

So is the conjecture folklore? Or can anybody point me to an explicit reference?

EDIT: Question has been up for a few days without answer so I'm giving up and assigning the result to "folklore". In the meantime, I've written a blog post about it for posterity: http://symomega.wordpress.com/2011/06/03/an-elusive-conjecture/

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    $\begingroup$ Sorry, nothing really helpful to say here, except: You could ask Kantor, he is still active, see pages.uoregon.edu/kantor You could also ask Nick Gill, who did some recent work on this subject, so he might know. Indeed, in this talk announcement, he claims that the conjecture goes back to Ostrom and Wagner: <win.tue.nl/seminar/abstracts/gill170506.pdf> But I could not find any further evidence for that; in particular, I don't see a mention of this in their original paper. But they could have made the conjecture elsewhere, of course. $\endgroup$
    – Max Horn
    Jun 1, 2011 at 11:22
  • $\begingroup$ @Max - I did actually ask Nick.. he did think that Ostrom & Wagner had made the conjecture in their 2-transitive paper; but their question/conjecture was actually about translation planes. I'll try Bill Kantor directly next.. $\endgroup$
    – Gordon Royle
    Jun 2, 2011 at 0:06
  • $\begingroup$ @GordonRoyle, I've recently come across the following paper of Koen Thas and Don Zagier (people.mpim-bonn.mpg.de/zagier/files/doi/10.4171/JEMS/107/…) where they say it was first mentioned in the Higman-McLaughlin's paper. I am wondering what is the status of your question today. $\endgroup$
    – Thibaut Dumont
    Apr 2, 2019 at 9:30
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    $\begingroup$ @ThibautDumont I don’t know anything more about this than when I originally asked, sorry! $\endgroup$
    – Gordon Royle
    Apr 3, 2019 at 10:42

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