International Journal of Mathematics and Mathematical Sciences
Volume 7 (1984), Issue 2, Pages 319-326

Quasifields with irreducible nuclei

Michael J. Kallaher

Department of Pure and Applied Mathematics, Washington State University, Pullman 99164-2930, Washington, USA

Received 22 January 1984

Copyright © 1984 Michael J. Kallaher. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 Rp or type Fp, 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.