(FUNDAMENTAL AND APPLIED MATHEMATICS)

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

## Some $2$-properties of the autotopism group of a $p$-primitive semifield plane

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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