computation of the order of GL(n,π½q)
GL(n,π½q) is the group of invertible nΓn matrices
over the finite field
π½q.
Here is a proof that
|GL(n,π½q)|=(qn-1)(qn-q)β―(qn-qn-1).
Each element AβGL(n,π½q) is given by a collection of n
π½q-linearly independent vectors (http://planetmath.org/LinearIndependence).
If one chooses the first column vector
of A from (π½q)n
there are qn choices, but one canβt choose the zero vector
since this would make the determinant
of A zero.
So there are really only qn-1 choices.
To choose an i-th vector from (π½q)n
which is linearly independent from i-1 already chosen
linearly independent vectors {V1,β¦,Vi-1}
one must choose a vector not in
the span of {V1,β¦,Vi-1}.
There are qi-1 vectors in this span,
so the number of choices is qn-qi-1.
Thus the number of linearly independent collections of n vectors in π½q
is (qn-1)(qn-q)β―(qn-qn-1).
Title | computation of the order of GL(n,π½q) |
---|---|
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 |