Associative Subalgebras of the Griess Algebra Original Research Article

The structure of the Griess algebra image, whose automorphism group is the Fischer-Griess monster F1 is investigated. We show the existence of 48-dimensional associative subalgebras in image and, furthermore, demonstrate that they are not contained in strictly larger ones. This is an immediate consequence of a more general necessary and sufficient criterion for the maximality of associative subalgebras of image. It is conjectured that the given explicit examples are of maximal possible dimension among all associative subalgebras in image. This depends on the validity of a certain inequality.