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Generators and Relations in Groups and Geometries pp 1–45Cite as

Classical Groups

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Part of the book series: NATO ASI Series ((ASIC,volume 333))

Abstract

If a group has a set of generators with outstanding properties, then the factorization of group elements into generators will provide information on the structure of the group. It is advantageous to determine the minimal number of factors needed to express an element as a product of generators. This number is called the length of a group element. The Cartan-Dieudonné theorem is a well-known example for results of this kind.

The classical groups have distinguished sets of generators. The general linear group is generated by simple mappings, the orthogonal group by reflections, the symplectic group by transvections, the unitary group by quasireflections, the group of projectivities by dilatations, the group of equiaffinities by translations and shears. The orthogonal group yields a second outstanding set of generators, namely the set of all orthogonal involutions.

We shall report on the solution of the length problem for a number of classical groups. We shall discuss whenever possible different generating sets and the resulting difference in the length of an element.

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  1. Department of Mathematics, University of Toronto, Toronto, Ontario, Canada, M5S 1A1

    Erich W. Ellers

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  1. Dipartimento di Matematica “Ulisse Dini”, Università degli Studi di Firenze, Florence, Italy

    A. Barlotti

  2. Department of Mathematics, University of Toronto, Toronto, Ontario, Canada

    E. W. Ellers

  3. Mathematisches Institut, Universität Erlangen-Nürnberg, Erlangen, Germany

    P. Plaumann  & K. Strambach  & 

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Ellers, E.W. (1991). Classical Groups. In: Barlotti, A., Ellers, E.W., Plaumann, P., Strambach, K. (eds) Generators and Relations in Groups and Geometries. NATO ASI Series, vol 333. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3382-1_1

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