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

I. V. Busarkina

Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 3, pp. 637-645

Résumé

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.

Bibliographie

[1] Podufalov N. D., Busarkina I. V., O $p$-primitivnykh polupolevykh ploskostyakh

[2] Busarkina I. V., Nekotorye $2$-svoistva gruppy avtotopizmov $p$-primitivnoi polupolevoi ploskosti