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The Monster and its nonassociative algebra. (English) Zbl 0582.20007

Finite groups - coming of age, Proc. CMS Conf., Montreal/Que. 1982, Contemp. Math. 45, 121-157 (1985).
[For the entire collection see Zbl 0565.00006.]
This article is intended to be an elementary exposition of the basic ingredients in the construction of the author’s Monster group, G (which he prefers to call the Friendly Giant). G is obtained as a very large subgroup of the group G(B), an important group associated with the algebra B. B is a commutative nonassociative algebra over the rational field of dimension 196,884 with a symmetric bilinear form (, ) which satisfies \((xy,z)=(x,yz)\) for all x,y,z in B. G(B) is the subgroup of the group of isometries of the form which preserves the algebra product. Quite fundamental to the construction are the Leech lattice and extraspecial p-groups, and the author starts by giving neat, succinct descriptions of these concepts.
The section headings indicate the pattern of the article: 1. Introduction. 2. The Leech lattice. 3. Extraspecial groups. 4. The C- module B. 5. G-algebras and retractions. 6. Retraction subalgebras of B. 7. The product of B and the definition of \(\sigma\). 8. Axioms for B? 9. Proof that \(\sigma\) is an algebra automorphism. 10. The identification of G. 11. Subgroups. 12. The pariahs and variants of this construction. 13. Update. Appended are three sections: A.1. Representations of extraspecial groups and their holomorphs. A.2. Simple subalgebras of B. A.3. Identities for B.
Reviewer: W.E.Deskins

MSC:

20D08 Simple groups: sporadic groups
17D99 Other nonassociative rings and algebras
17A99 General nonassociative rings

Citations:

Zbl 0565.00006

ATLAS of Finite Group Representations:

Monster group M