If a finite projective plane $\pi_1$ of order $m$ contains, as a sub plane, a finite projective plane $\pi_2$ of order $n$, then $m \geq n^2$ with equality holding only in the case of a Baer sub plane. Otherwise $m \geq n^2 + n$. (This is a theorem of Bruck that can be found in Hall's Group Theory book, I believe.) My question is, how nearly can this bound be achieved?

For example, we have cases for $m=9$ and $n=2$, where $\pi_1$ is one of the non-Desaurgian planes of order $9$, and $\pi_2$ is the Fano plane. Is anything known about how close we can come to the bound in the general case? Has the bound been improved?

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    $\begingroup$ We are trying to implement designs constructions in Sage. We did a lot of work for OA/TD/MOLS already but designs are all great and there is too much stuff to implement. If you like combinatorial designs and code send me an email (I didn't find yours in your profile) there is a lot to do ! $\endgroup$ Jul 28, 2014 at 15:55
  • $\begingroup$ Not sure about the bound in general but there are some results about existence of sub-planes that are known. For example, combinatorics.org/ojs/index.php/eljc/article/view/v18i1p2. For details on small planes you can also refer to uwyo.edu/moorhouse/pub/planes $\endgroup$
    – Anurag
    Jan 12, 2015 at 12:48
  • $\begingroup$ In general an algebraic projective plane of order $p^e$ contains algebraic subplanes of order $p^f$ for all factors $f|e$, so in particular $n=2$ is seen already for $m=8$. I think it's known that the only projective plane of order $7$ is algebraic, and that one does not contain the Fano plane. $\endgroup$ Apr 23, 2019 at 2:38

1 Answer 1


No, the bound has not been improved. Even a subplane of order 3 of a projective plane of order 12 has not yet been ruled out. (You can see this in the arguments in the work on possible collneations of a projective plane of order 12, such as Janko-Van Trung (1980) [On projective planes of order 12 which have a subplane of order 3. I. J. Combin. Theory Ser. A 29, 254–256].) There's a result of Roth from 1964 that if the subplane is fixed by a non-trivial collineation then $n > m^2+m+1$ [Roth, Richard (1964). Collineation groups of finite projective planes. Mathematische Zeitschrift, 83(5), 409-421, Corollary 2 on page 418]. Of course, if you found a subplane with equality in the Bruck bound, you would have a projective plane of order not a prime power, and have solved a much bigger problem, by disproving the prime power conjecture for finite projective planes.


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