elementary proof of orders
When possible, our proofs avoid matrices so that the proofs retain some value to infinite dimensional settings. When we use we mean any field, and indicates the special case of a finite field of order . is always our vector space.
There are many alternative methods for computing orders of classical groups, for instance observing special subgroups (http://planetmath.org/TheoryFromOrdersOfClassicalGroups2) or from Lie theory and the study of Chevalley groups. The method explored here is intended to be elementary linear algebra.
The basic starting point in computing orders of classical groups is an application of elementary linear algebra rephrased in group theory terms.
acts regularly on the set of ordered bases of vector space over a field .
Given any two bases and of a vector space , define the map by
[Note the above sum has only finitely many non-zero .] It follows is an invertible linear transformation so . Furthermore, any linear transformation with must satisfy (1) to be linear so indeed . Therefore acts regularly on ordered bases of . ∎
In the world of group theory, a regular action is a typical substitute for knowing the order of a group. In particular, any two groups, even infinite, have the same order if they have a regular action on the same set. However, we are presently after specific order of finite groups so we return to the case of a finite dimension vector space over a finite field . We do however attempt to establish the orders through bijections with other sets and groups so that the results apply in more general contexts as well.
When , the number of ordered bases can be counted. A basis is a set of linearly independent vectors. So may be chosen freely from , providing possible choices. Next must be chosen independent form so can be freely chosen from leaving choices. In a similar fashion has possiblities and so continuing by induction we find the total number of ordered bases to be:
As acts regularly on ordered bases of , this is the order of .
so that .
In a similar process, so if we derive the order of the center of we derive the order of . The central transforms are scalar (they must preserve every eigenspace of every linear transform) so is isomorphic to . Thus the order of is the same as the order of .
For and simply notice so when we get an additional term.
Finally, we consider . The order of must be computed. So we require scalar transforms with determinant 1. As the such, if is the eigen value of the scalar transform we need in . From finite field theory we know . As this group is cyclic we know that every element satisfying also satisfies and lies in the unique subgroup of order of . Thus . ∎
|Title||elementary proof of orders|
|Date of creation||2013-03-22 15:56:52|
|Last modified on||2013-03-22 15:56:52|
|Last modified by||Algeboy (12884)|