chapter  11
Linear Groups and Groups of Isometries
Pages 62

In this chapter we study certain subgroups of the group of units GL(V ) in the algebra L(V, V ) where V is an n-dimensional vector space over a field F. In the first section we consider the normal group SL(V ) of GL(V ) consisting of those operators of determinant 1. We show that except when (n,F) = (2,F2) or (3,F3), this group is perfect, and then prove that the quotient group of SL(V ) by its center is a simple group. In the second section we equip V with a non-degenerate alternating bilinear form f and study the group I(V, f) of isometries f . Section three is devoted to isometries of a non-degenerate orthogonal space over a field F where the characteristic of F is not two. The final section is concerned with groups of isometries of a finite-dimensional, non-degenerate unitary space.