Linear classical group

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).