Frobenius groups: it's all algebraic number theory
by
Ron Brown
University of Hawaii

Frobenius groups play a prominent role in the general theory of finite groups. After reviewing definitions we show that the problem of finding all Frobenius groups with abelian Frobenius kernel is a problem in algebraic number theory. We illustrate this with a very elementary construction of all Frobenius groups whose Frobenius complement is either the quaternion group or the special linear group SL(2,Z/5Z) and with some combinatorial results (e.g., a count of the metabelian Frobenius groups of order at most one million). The key idea is that Frobenius groups with abelian kernel correspond precisely to finite modules over certain well-behaved classical maximal orders, and the indecomposable modules correspond to powers of maximal ideals of the associated Dedekind domain.

Connections with Amitsur's calculation of the finite groups which are subgroups of division rings will be mentionned.

Date received: February 9, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacv-07.