## Bulletin of the Belgian Mathematical Society - Simon Stevin

### The dimension of a subplane of a translation plane

#### Abstract

It is shown that the commutative binary Knuth semifield planes of order $2^{n}$, for $n=5k~$\ or $7k$ and $k$ odd, their transposes and transpose-duals admit subplanes of order $2^{2}$. In addition, many of the Kantor commutative semifield planes of order $2^{5k}$ or $2^{7k}$ also admit subplanes of order $4$. Furthermore, a large number of maximal partial spreads of order $p^{k}$ and deficiency at least $p^{k}-p^{k-1}$ or translation planes of order $p^{k}$ are constructed using direct sums of matrix spreads sets of different dimensions. Given any translation plane $\pi_{0}$ of order $p^{d}$, there is either a proper maximal partial spread of order $p^{c+d}$ whose associated translation net contains a subplane of order $p^{d}$ isomorphic to $\pi_{0}$ or there is a translation plane of order $p^{c+d}$ admitting a subplane of order $p^{d}$. Other than the semifield planes mentioned above and a few sporadic planes of even order, there are no other known translation planes of order $p^{c+d}$ admitting a subplane of order $p^{d}$, where $d$ does not divide $c$.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 17, Number 3 (2010), 463-477.

Dates
First available in Project Euclid: 15 September 2010

https://projecteuclid.org/euclid.bbms/1284570732

Digital Object Identifier
doi:10.36045/bbms/1284570732

Mathematical Reviews number (MathSciNet)
MR2731368

Zentralblatt MATH identifier
1213.51003

#### Citation

Jha, V.; Johnson, N.L. The dimension of a subplane of a translation plane. Bull. Belg. Math. Soc. Simon Stevin 17 (2010), no. 3, 463--477. doi:10.36045/bbms/1284570732. https://projecteuclid.org/euclid.bbms/1284570732