elementary proof of orders

When possible, our proofs avoid matrices so that the proofs retain some value to infinite dimensional settings. When we use k we mean any field, and GF(q) indicates the special case of a finite field of order q. V is always our vector spaceMathworldPlanetmath.

Remark 1.

There are many alternative methods for computing orders of classical groupsMathworldPlanetmath, for instance observing special subgroupsMathworldPlanetmathPlanetmath (http://planetmath.org/TheoryFromOrdersOfClassicalGroups2) or from Lie theory and the study of Chevalley groups. The method explored here is intended to be elementary linear algebraMathworldPlanetmath.

The basic starting point in computing orders of classical groups is an application of elementary linear algebra rephrased in group theory terms.

Proposition 2.

GL(V) acts regularly on the set of ordered bases of vector space V over a field k.


Given any two bases B={bi:iI} and C={ci:iI} of a vector space V, define the map f:VV by

(iIlibi)f=iIlici. (1)

[Note the above sum has only finitely many non-zero lik.] It follows f is an invertible linear transformation so fGL(V). Furthermore, any linear transformation gGL(V) with (bi)g=ci must satisfy (1) to be linear so indeed g=f. Therefore GL(V) acts regularly on ordered bases of V. ∎

In the world of group theory, a regularPlanetmathPlanetmathPlanetmath action is a typical substitute for knowing the order of a group. In particular, any two groups, even infiniteMathworldPlanetmath, have the same order if they have a regular action on the same set. However, we are presently after specific order of finite groupsMathworldPlanetmath so we return to the case of V a finite dimensionPlanetmathPlanetmath vector space over a finite field k=GF(q). 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.

Theorem 3.

When V=GF(q)d, the number of ordered bases can be counted. A basis is a set B={b1,,bd} of linearly independentMathworldPlanetmath vectors. So b1 may be chosen freely from V-{0}, providing qd-1 possible choices. Next b2 must be chosen independent form b1 so b2 can be freely chosen from V-b1 leaving qd-q choices. In a similarPlanetmathPlanetmath fashion b3 has qd-q2 possiblities and so continuing by inductionMathworldPlanetmath we find the total number of ordered bases to be:


Now we treat q as the variable of a polynomial and factor this number into:


As GL(V) acts regularly on ordered bases of V, this is the order of GL(V).

For the order of SL(V) recall the SL(V) is the kernel of the determinantMathworldPlanetmath homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath det:GL(V)k×. Furthermore, det is surjectivePlanetmathPlanetmath as a diagonal matrixMathworldPlanetmath can be used to exhibit any determinant we seek. We conclude


so that |SL(d,q)|=|GL(d,q)|/(q-1).

In a similar process, PGL(V)=GL(V)/Z(GL(V)) so if we derive the order of the center of GL(V) we derive the order of PGL(V). The central transforms are scalar (they must preserve every eigenspaceMathworldPlanetmath of every linear transform) so Z(GL(V)) is isomorphicPlanetmathPlanetmath to k×. Thus the order of PGL(V) is the same as the order of SL(V).

For ΓL(V) and PΓL(V) simply notice ΓL(V)=GL(V)k× so when k=GF(q) we get an additional q-1 term.

Finally, we consider PSL(V)=SL(V)/(SL(V)Z(GL(V))). The order of SL(V)Z(GL(V)) must be computed. So we require scalar transforms with determinant 1. As the such, if r is the eigen value of the scalar transform we need rd=1 in GF(q). From finite field theory we know GF(q)×q-1. As this group is cyclic we know that every element rGF(q)× satisfying rd=1 also satisfies rq-1=1 and r lies in the unique subgroup of order (d,q-1) of GF(q)×. Thus |SL(V)Z(GL(V))|=(d,q-1). ∎

Title elementary proof of orders
Canonical name ElementaryProofOfOrders
Date of creation 2013-03-22 15:56:52
Last modified on 2013-03-22 15:56:52
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 8
Author Algeboy (12884)
Entry type Proof
Classification msc 11E57
Classification msc 05E15