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(2,3)‐Generation of Exceptional Groups
Author(s) -
Lübeck Frank,
Malle Gunter
Publication year - 1999
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s002461079800670x
Subject(s) - mathematics , classification of finite simple groups , lie group , pure mathematics , abelian group , type (biology) , simple lie group , group of lie type , simple (philosophy) , rank (graph theory) , simple group , field (mathematics) , finite field , group (periodic table) , combinatorics , algebra over a field , group theory , physics , ecology , philosophy , epistemology , quantum mechanics , biology
We study two aspects of generation of large exceptional groups of Lie type. First we show that any finite exceptional group of Lie rank at least four is (2,3)‐generated, that is, a factor group of the modular group PSL 2 (Z). This completes the study of (2,3)‐generation of groups of Lie type. Second, we complete the proof that groups of type E 7 and E 8 over fields of odd characteristic occur as Galois groups of geometric extensions of Q ab ( t ), where Q ab denotes the maximal Abelian extension field of Q. Finally, we show that all finite simple exceptional groups of Lie type have a pair of strongly orthogonal classes. The methods of proof in all three cases are very similar and require the Lusztig theory of characters of reductive groups over finite fields as well as the classification of finite simple groups.