Abstract
In this note, I announce the construction of the finite simple group F1, whose existence was predicted independently in 1973 by Bernd Fischer and by me. The group has order 246320597611213317.19.23.29.31.41. 47.59.71 = 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 and is realized as a group of automorphisms of a 196,883-dimensional commutative nonassociative algebra over the rational numbers, which has an associative form. Equivalently, it is a group of automorphisms of a cubic form in 196,883 variables. It turns out that all the relevant arguments and calculations may be done by hand. Furthermore, existence of the group F1 implies the existence of a number of other sporadic simple groups for which existence proofs formerly depended on work with computers. We are beginning to look upon this group as a “friendly giant.”
Keywords: Fischer—Griess simple group, sporadic simple group, classification of finite simple groups, commutative nonassociative algebra, Leech lattice
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