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2-Generation of finite simple groups and some related topics

  • L. Di Martino
  • M. C. Tamburini
Chapter
Part of the NATO ASI Series book series (ASIC, volume 333)

Abstract

It is well-known that every finite simple group is 2-generated, i.e. it can be generated by two suitable elements. This is the topic discussed in §1, which centers around Steinberg’s unified treatment of groups of Lie type. In §2 we discuss generation of simple groups by special kinds of generating pairs, namely: 1) the generation of simple groups of Lie type by a cyclic maximal torus and a long root element, with application to the solution of the Magnus-Gorchakov-Levchuk conjecture on residual properties of free groups; 2) the generation of a simple group by an involution and another suitable element. With regard to 1), we also mention similar 2-generation results in connection with Galois groups; with regard to 2), emphasis is put on (2,3)-generation and Hurwitz generation of finite simple groups. Finally, §3 deals with generating sets of involutions of minimal size. Most finite simple groups are generated by three involutions. Generation results, a non-generation criterion, and a relation between (2,3)-generation and generation by three involutions are illustrated.

Keywords

Finite Group Simple Group Maximal Subgroup Weyl Group Galois Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media Dordrecht 1991

Authors and Affiliations

  • L. Di Martino
    • 1
  • M. C. Tamburini
    • 2
  1. 1.Dipartimento di Matematica „F. Enriques“Università degli Studi di MilanoMilanoItaly
  2. 2.Dipartimento di MatematicaUniversità Cattolica del Sacro CuoreBresciaItaly

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