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Presentations of some finite simple groups. (English) Zbl 0663.20014

The author gives presentations of the sporadic simple groups \(M_{11}\), \(M_{12}\), \(M_{22}\), \(M_{23}\), \(M_{24}\), \(J_ 1\), \(J_ 2\), HS, McL, and \(Co_ 3\). The generators in these presentations are involutions. If \(<S:R>\) is a presentation of the group G and l(r) is the length of the relator r as an element of the free group over S, then one defines \(| R| +\sum_{r\in R}l(r)\) as the length of the presentation. In this sense the presentations given by the author are shorter than the presentations of these groups in the ATLAS [Atlas of finite groups, J. H. Conway et al. (Clarendon Press, Oxford 1985; Zbl 0568.20001)]. The presentations were found and verified with the help of a computer. The article includes a discussion of the computational aspects and the essential part, the language CAYLEY [see J. Cannon, in Computational Group Theory, Durham 1982, 145-183 (1984; Zbl 0544.20002)] plays in the proofs.
Reviewer: U.Dempwolff

MSC:

20D08 Simple groups: sporadic groups
20F05 Generators, relations, and presentations of groups
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