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Generation of exceptional groups of Lie-type. (English) Zbl 0758.20001
This paper is concerned with finding ways of generating a finite group of Lie type in one of the ten ‘exceptional’ families, with a smaller number of elements. Theorem A shows how to choose a particular semisimple element \(s\) in such a group \(G\), such that \(\langle s\rangle\) is a maximal torus, and with the property that ‘almost all’ elements \(x\) in \(G\) generate \(G\) (i.e. \(\langle s,x\rangle=G\)). This is proved by looking at the list of maximal subgroups of \(G\), or if that is not available, the list of maximal subgroups containing \(s\), and estimating the number of elements \(x\) such that \(\langle s,x\rangle\) is a proper subgroup. Theorem \(B\) is a corollary, stating that, except in the cases \(E_ 6\) and \(^ 2E_ 6\), every finite simple exceptional group of Lie type can be generated by three involutions. This uses the fact that in most cases \(s\) has been chosen to have odd order, to show that \(s\) is the product of two involutions, and \(x\) can also be chosen to be an involution. Whilst it appears that most finite simple groups can be generated by three involutions, it is worth noting that \(U_ 3(3)\) can not.

MSC:
20D06 Simple groups: alternating groups and groups of Lie type
20F05 Generators, relations, and presentations of groups
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