A construction of F 1 as automorphisms of a 196,883-dimensional algebra
- 1 February 1981
- journal article
- Published by Proceedings of the National Academy of Sciences in Proceedings of the National Academy of Sciences
- Vol. 78 (2) , 689-691
- https://doi.org/10.1073/pnas.78.2.689
Abstract
In this note, I announce the construction of the finite simple group F(1), whose existence was predicted independently in 1973 by Bernd Fischer and by me. The group has order 2(46)3(20)5(9)7(6)11(2)13(3)17.19.23.29.31.41. 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 F(1) 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
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