special linear group

English

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 SL_n(F) — or, if the field is understood, SL(n) or SL_n. 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.

Derived terms

• projective special linear group

Translations

group of n×n invertible matrices with determinant 1

See also

• general linear group
• Lie group

Further reading

• Modular group on Wikipedia.Wikipedia
• Projective linear group on Wikipedia.Wikipedia
• SL2(C) on Wikipedia.Wikipedia
• SL2(R) on Wikipedia.Wikipedia