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Generators for finite simple Moufang loops
by
Petr Vojtěchovský
Iowa State University
Recall that loop is a quasigroup with an identity element, i.e. a non-associative ``group''. The variety of Moufang loops, characterized by the near-associativity law x(y(zy))=((xy)z)y), exhibits many properties known to group theorists.
The class of finite simple Moufang loops M was described by M. Liebeck in 1987. It turned out that every member of M is either a finite simple group or a multiplicative subloop of one of the so-called Cayley-Dickson algebras. The latter, infinite class of loops was constructed by L. Paige in 1956.
It is one of the remarkable consequences of the classification of finite simple groups that every finite simple group happens to be two-generated. This cannot be the case for non-associative Moufang loops, as shown by Bruck, and it is thus natural to expect that three generators will do.
Indeed, we use Dickson's result on generators of SLq(2) to prove that every non-associative finite simple Moufang loop (with one possible exception) is three-generated.
Date received: May 29, 2000
Copyright © 2000 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 # caee-80.