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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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