# Subplanes of Finite Projective Planes

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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• 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 Jan 12, 2015 at 12:48
• 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. Apr 23, 2019 at 2:38

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.