computation of the order of
is the group of invertible matrices
over the finite field
.
Here is a proof that
.
Each element is given by a collection of
-linearly independent vectors (http://planetmath.org/LinearIndependence).
If one chooses the first column vector
of from
there are choices, but one canβt choose the zero vector
since this would make the determinant
of zero.
So there are really only choices.
To choose an -th vector from
which is linearly independent from already chosen
linearly independent vectors
one must choose a vector not in
the span of .
There are vectors in this span,
so the number of choices is .
Thus the number of linearly independent collections of vectors in
is .
Title | computation of the order of |
---|---|
Canonical name | ComputationOfTheOrderOfoperatornameGLnmathbbFq |
Date of creation | 2013-03-22 13:06:50 |
Last modified on | 2013-03-22 13:06:50 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 15 |
Author | yark (2760) |
Entry type | Proof |
Classification | msc 20G15 |
Related topic | OrderOfTheGeneralLinearGroupOverAFiniteField |