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

I. V. Busarkina


Abstract: 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.

UDC: 519.145

Received: 01.05.1998

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