## Bulletin of the Belgian Mathematical Society - Simon Stevin

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

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

V. Jha and N.L. Johnson

#### 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

**Permanent link to this document**

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