special linear group

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English

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Noun

special linear group (plural special linear groups)

  1. (group theory) For given field F and order n, the group of n×n matrices with determinant 1, with the group operations of matrix multiplication and matrix inversion.
    • 1972, Johan G. F. Belinfante, Bernard Kolman, A Survey of Lie Groups and Lie Algebras with Applications and Computational Methods, Society for Industrial and Applied Mathematics, page 6:
      The complex special linear group SL(n, C) is the subgroup of GL(n, C) consisting of matrices with determinant one. […] The special linear groups are sometimes also called unimodular groups.
    • 1998, F. Celler, C. R. Leedham-Green, A constructive recognition algorithm for the special linear group, Robert Curtis, Robert Wilson (editors), The Atlas of Finite Groups: Ten Years On, Cambridge University Press, 2003 Digitally Printed Edition, page 11,
      In the first part of this note we present an algorithm to recognise constructively the special linear group.
    • 2014, Holger Ingmar Meinhardt, The Pre-Kernel as a Tractable Solution for Cooperative Games: An Exercise in Algorithmic Game Theory, Springer, page 100:
      Hence, all congruent bases induced from a game context belong to the special linear group SL(m).

Usage notes

The special linear group can be denoted SL(n, F) or SLn(F) — or, if the field is understood, SL(n) or SLn. It is a normal subgroup of the general linear group GL(n,F). In the cases that F is the field of real or of complex numbers, SL(n, F) is a Lie group.

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