Abstract

This article considers finite quasifields having a subgroup N of either the right or middle nucleus of Q which acts irreducibly as a group of linear transformations on Q as a vector space over its kernel. It is shown that Q is a generalized André system, an irregular nearfield, a Lüneburg exceptional quasifield of type R∗p or type F∗p, or one of four other possibilities having order 52, 52, 72, or 112, respectively. This result generalizes earlier work of Lüneburg and Ostrom characterizing generalized André systems, and it demonstrates the close similarity of the Lüneburg exceptional quasifields to the generalized André system.