I have the statement 
in the book " 3-d transposition groups" by Michael Aschbacher. What does this mean?
Does this mean that the general linear group can be regarded as the isometry group of a null vector space?
I am struggling to find anything that mentions a bilinear form for the general linear group.
Many thanks.
given a vector space with a fixed basis, a quadratic form is given by $v^T H v,$ where $H$ is symmetric and $v$ a column vector and $v^T H v$ a 1 by 1 matrix, i.e. a number.
next, an isometry is a nonsingular linear transformation, take it as an invertible matrix $P$, such that $$ P^T HP = H. $$
Finally, if $H$ is the zero matrix, then $P$ can be any invertible matrix, as $P^T 0 P = 0$