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Linear classical group

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A group of non-singular linear transformations of a finite-dimensional vector space $ E $ over a skew-field $ K $ that is a classical group (see also Linear group). The most important types of linear classical groups are the following: the general linear group $ \mathop{\rm GL} _ {n} ( K) $, the special linear group $ \mathop{\rm SL} _ {n} ( K) $ and the unitary group $ U _ {n} ( K , f ) $( where $ n = \mathop{\rm dim} E $ and $ f $ is a Hermitian or skew-Hermitian form on $ E $, relative to an involution of $ K $). When $ K $ is also commutative, special important cases are: the symplectic group $ \mathop{\rm Sp} _ {n} ( K) $ and the orthogonal group $ O _ {n} ( K , f ) $( $ f $ a quadratic form on $ E $ and $ K $ of characteristic not 2).

How to Cite This Entry:
Linear classical group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_classical_group&oldid=47649
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article