On primitive subgroups of full affine groups of finite semi-fields

K. K. Shchukin

Diskretnaya Matematika, Tome 15 (2003) no. 4, pp. 126-132

Résumé

In this paper, we continue and complete the study of finite primitive groups whose stabiliser of a point contains an Abelian normal subgroup acting irreducibly (by conjugations) on an Abelian normal subgroup of the whole group. Each such group $H$ is isomorphic to the subgroup $Z_p^\nu \leftthreetimes\Theta \leftthreetimes\Psi$ of the full affine group $A(F_{p^\nu})\cong Z_p^\nu \leftthreetimes Z_{p^\nu-1} \leftthreetimes Z_p$ of the field $F_{p^\nu}$, where the symbol of the semi-direct product $\leftthreetimes$ unites the $\nu$-power of the cyclic group $Z_p$, the metacyclic group $\Theta$, and some group of automorphisms $\Psi$ of the field $F_{p^\nu}$. Using the Zassenhaus classification of finite semi-fields, we enumerate primitive subgroups of the full affine groups of finite semi-fields.