FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2002, VOLUME 8, NUMBER 3, PAGES 637-645

I. V. Busarkina

Abstract

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```
Let
```$\pi$ be a semifield plane of order $q^4$
with the regular set

$$

\Sigma = \left\{

\begin{bmatrix}

u & \tau v\\
f(v) & u^q

\end{bmatrix}

\;\biggm|\; u,v,f(v) \in GF(q^2)=F
\right\},

$$

$f(v)=f_0v+f_1v^p+\ldots+f_{2r-1}v^{p^{2r-1}}$
be an additive function on $F$ , $\tau$ normalize the field,
$q=p^r$ and $p>2$ be a prime number.
If the plane has rank $4$ and $f(v)=f_0v$ or $f(v)=f_rv^q$ ,
then the $2$ -rank of the autotopism group is $3$ and some Sylow
$2$ -subgroup $S$ of the group $A$ has the form
$S=H_2\cdot\langle g\rangle\langle g_1\rangle$ ,
where $H_2$ is a Sylow $2$ -subgroup of the group $H$ ,
and $g$ , $g_1$ are $2$ -elements of a certain form.

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Last modified: February 17, 2003