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\((2,3)\)-generation of exceptional groups. (English) Zbl 0935.20021
A group \(G\) is called \((2,3)\)-generated if there are \(x,y\in G\) with \(O(x)=2\), \(O(y)=3\) such that \(G=\langle x,y\rangle\). In this case \(G\) is isomorphic to a homomorphic image of the modular group \(\text{PSL}_2(\mathbb{Z})\). In previous works [G. Malle, Can. Math. Bull. 33, No. 3, 349-357 (1990; Zbl 0734.20013) and Lond. Math. Soc. Lect. Note Ser. 207, 173-183 (1995; Zbl 0861.20017)] it was shown that the finite simple exceptional groups \(^2G_2(q)\), \(G_2(q)\), \(^3D_4(q)\) and \(^2F_4(q)\) are \((2,3)\)-generated. In the paper under review, the authors prove that the simple groups of type \(F_4(q)\), \(E_6(q)\), \(^2E_6(q)\), \(E_7(q)\) and \(E_8(q)\) are all \((2,3)\)-generated. Therefore it is proved that all the finite simple exceptional groups of Lie type, apart from \(^2B_2(2^{2n+1})\), are \((2,3)\)-generated. The authors also complete the proof of the following fact that any finite simple exceptional group of Lie type in odd characteristic occurs as the Galois group of a field extension of \(\mathbb{Q}^{\text{ab}}(t)\), where \(\mathbb{Q}^{\text{ab}}\) denotes the maximal Abelian extension field of \(\mathbb{Q}\). Finally, it is proved that for any finite simple exceptional group of Lie type \(G\) there is a conjugacy class \(C\) such that \(G\) is covered by \(C^2\cup C^3\), i.e. \(G=C^2\cup C^3\).

20F05 Generators, relations, and presentations of groups
20D06 Simple groups: alternating groups and groups of Lie type
12F12 Inverse Galois theory
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