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**A simple construction for the Fischer-Griess monster group.**
*(English)*
Zbl 0564.20010

In [Invent. Math. 69, 1–102 (1982; Zbl 0498.20013)] R. Griess gave the construction of the largest, sporadic finite simple group \(F\), the Fischer-Griess monster, as the group of automorphisms of a certain 196884-dimensional algebra \(B\) in a Euclidean space. It came as a surprise that such a large group was constructed essentially without a computer. Still the construction of Griess contained extremely difficult technical details. In [ibid. 78, 491–499 (1984; Zbl 0548.20011)] and some other papers J. Tits gave a variation of the construction of Griess. This made many computations obsolete and showed in particular in a simple way, that \(\operatorname{Aut}(B)\) is a finite group.

In the paper under review the author gives a further variation of the construction of \(B\) and \(F\). His aim is to make computations in the algebra \(B\) as easy as possible. Griess constructs \(F\) by the centralizer \(C\) of an involution and the normalizer \(N\) of a four group. He starts with a representation of \(C\), defines a \(C\)-invariant algebra \(B\), and finds an involution \(\sigma\in N-C\), and works out in a very delicate analysis the precise shape of \(\sigma\). The author of the paper under review goes the opposite way: In a natural way he finds a fourfold cover of \(N\) as the automorphism group of a certain Moufang loop. He then works from \(N\) to \(C\). The convenient presentation of \(N\) pays off in a simplification of computations and formulas which occur when \(C\) and \(N\) are “put together”. With the facilitation of the computations in the algebra \(B\) one also can show, that there is a certain finite number of vectors, which is invariant under \(F=\operatorname{Aut}(B)\), proving the finiteness of \(F\). Finally the author gives various results on subalgebras of \(B\) including some work of S. Norton.

In the paper under review the author gives a further variation of the construction of \(B\) and \(F\). His aim is to make computations in the algebra \(B\) as easy as possible. Griess constructs \(F\) by the centralizer \(C\) of an involution and the normalizer \(N\) of a four group. He starts with a representation of \(C\), defines a \(C\)-invariant algebra \(B\), and finds an involution \(\sigma\in N-C\), and works out in a very delicate analysis the precise shape of \(\sigma\). The author of the paper under review goes the opposite way: In a natural way he finds a fourfold cover of \(N\) as the automorphism group of a certain Moufang loop. He then works from \(N\) to \(C\). The convenient presentation of \(N\) pays off in a simplification of computations and formulas which occur when \(C\) and \(N\) are “put together”. With the facilitation of the computations in the algebra \(B\) one also can show, that there is a certain finite number of vectors, which is invariant under \(F=\operatorname{Aut}(B)\), proving the finiteness of \(F\). Finally the author gives various results on subalgebras of \(B\) including some work of S. Norton.

Reviewer: Ulrich Dempwolff (Kaiserslautern)

### Keywords:

sporadic finite simple group; Fischer-Griess-monster; group of automorphisms; 196884-dimensional algebra; cover; Moufang loop### References:

[1] | Griess, R.L.: The Friendly Giant. Invent. Math.69, 1-102 (1982) · Zbl 0498.20013 |

[2] | Parker, R.A., Conway, J.H.: A remarkable Moufang loop, with an application to the Fischer group Fi24. In press (1985) |

[3] | Conway, J.H.: Three lectures on exceptional groups. In: Finite Simple Groups. Higman, G., Powell, M. (eds.) London: Academic Press 1971 · Zbl 0242.20013 |

[4] | Tits, J.: Remarks on Griess’s construction of the Monster, I, II, III, IV. Preprint |

[5] | Frenkel, I., Lepowsky, J., Meurman, A.: A natural representation of the Fischer-Griess Monster with the modular functionj as character. Preprint, Mathematical Sciences Research Institute, Berkeley, California, 1984 · Zbl 0543.20016 |

[6] | Fischer, B., Livingstone, D., Thorne, M.: The characters of the Monster group. University of Birmingham, 1978 |

[7] | Griess, R.L.: The Monster and its Nonassociative Algebra. To appear in the proceedings of the conference: ?Finite Groups: Coming of Age?, Montreal 1982 |

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