The uniqueness of groups of type \(J_ 4\).

*(English)*Zbl 0735.20006From the introduction of the authors: “We give the first computer free proof of the uniqueness of groups of type \(J_ 4\). In addition we supply simplified proofs of some properties of such groups, such as the structure of certain subgroups.…Around 1980, Norton, Parker, and Thackray proved the existence and uniqueness of groups of type \(J_ 4\) using extensive machine computation.…Norton et al. constructed \(J_ 4\) as a linear group in 112 dimensions over the field of order 2. While the notion of a 2-local geometry did not exist at that time, this geometry plays an implicit role…”

The authors’ proof is based on the theory of so called uniqueness systems presented by them in a forthcoming paper. This theory is then applied to the 2-local geometry of a group \(G\) of type \(J_ 4\) and its collinearity graph. The main step in the uniqueness proof is to show that the graph is simply connected. Here, a group of type \(J_ 4\) is a simple group as (discovered and) described by Zvonimir Janko in 1976.

The authors’ proof is based on the theory of so called uniqueness systems presented by them in a forthcoming paper. This theory is then applied to the 2-local geometry of a group \(G\) of type \(J_ 4\) and its collinearity graph. The main step in the uniqueness proof is to show that the graph is simply connected. Here, a group of type \(J_ 4\) is a simple group as (discovered and) described by Zvonimir Janko in 1976.

Reviewer: D.Held

##### MSC:

20D08 | Simple groups: sporadic groups |

51D20 | Combinatorial geometries and geometric closure systems |

51E24 | Buildings and the geometry of diagrams |

05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |

##### Keywords:

computer free proof; groups of type \(J_ 4\); uniqueness systems; 2-local geometry; collinearity graph; uniqueness proof; simply connected; simple group
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\textit{M. Aschbacher} and \textit{Y. Segev}, Invent. Math. 105, No. 3, 589--607 (1991; Zbl 0735.20006)

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##### References:

[1] | Aschbacher, M.: Finite Group Theory. Cambridge: Cambridge University Press, 1986 · Zbl 0583.20001 |

[2] | Aschbacher, M.: The geometry of trilinear forms. In: W. Kantz (ed.), Finite Geometries, Buildings, and Related Topics, pp. 75-84, Oxford: Oxford University Press, 1990 · Zbl 0752.11020 |

[3] | Aschbacher, M., Segev, Y.: Extending morphisms of groups and graphs. Ann. Math. (to appear) · Zbl 0778.20009 |

[4] | Held, D.: The simple groups related toM 24. J. Algebra13, 253-296 (1969) · Zbl 0182.04302 |

[5] | James, G.: The modular characters of the Mathieu groups. J. Algebra27, 57-111 (1973) · Zbl 0268.20008 |

[6] | Janko, Z.: A new finite simple group of order 86, 775, 571, 046, 077, 562, 880 which possessesM 24 and the full cover ofM 22 as subgroups. J. Algebra42, 564-596 (1976) |

[7] | Norton, S.: The construction ofJ 4. Proc. Sym. Pure Math.37, 271-278 (1980) |

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