theory from orders of classical groups

One formulaMathworldPlanetmathPlanetmath for the order of GL(d,q) can be given as:


In Orders of Classical GroupsMathworldPlanetmath ( we see an accounting for this orders from an elementary linear algebraMathworldPlanetmath perspective and from there derive order of related classical groups. However, many of these formulas can be explicitly observed with the group theoretic structureMathworldPlanetmath of the various classical groups. We explore this presently.

We describe three families of subgroupsMathworldPlanetmathPlanetmath U, T, Si which intersect trivially and whose orders are:

|U|=q(d2),|T|=(q-1)d, and |Si|=qi-1++q+1,2id.

1 The Unipotent matrices U

Proposition 1.

Given q=ps a Sylow p-subgroup of GL(d,q) has order ps(d2) and is isomorphicPlanetmathPlanetmathPlanetmathPlanetmath to the group of lower-unitriangular matrices.


Observe for any i>0, qi-1-1(modq). Therefore qi-1-1(modp) as q=ps. Therefore p is relatively prime to the


factor of the order of GL(d,q). Therefore the order of a Sylow p-subgroup of GL(d,q) is


From matrix multiplicationMathworldPlanetmath we observe the that productPlanetmathPlanetmath of two lower triangular matricesMathworldPlanetmath is lower triangular. And the same for lower unitriangular. Therefore the lower unitriangular matrices form a subgroup of GL(d,k). Moreover, they have (d2) entries which can range over GF(q) giving a total order of q(d2).


Remark 2.

Unitriangular subgroups are often called unipotent because every element in the group has all eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath equal to 1.

Definition 3.

A unipotent subgroup of GL(V) is a subgroup in which every element has all eigenvalues equal to 1.

It can be shown that every unipotent group, under some choice of basis, is a set of lower unitriangular matrices. Of course upper triangular matrices may also be used.

2 The Diagonal (Toral) matrices T

We will now account for the (q-1)d term in the order.

Proposition 4.

The diagonal matricesMathworldPlanetmath of GL(d,k) form an abelian groupMathworldPlanetmath isomorphic to (k×)d. So in GL(d,q), the diagonal matrices have order (q-1)d.


The isomorphismPlanetmathPlanetmathPlanetmathPlanetmath is easily explained with matrix multiplication:


Note that in order to be invertiblePlanetmathPlanetmathPlanetmathPlanetmath each ai0. ∎

Remark 5.

In general theory, diagonal matrices are replaced with the term toral subgroup. This is on account of the fact that when k=C, C× is homotopic to the circle, S1. Therefore, as a topology, (C×)d is S1×S1××S1, that is, the d-dimensional torus.

Definition 6.

A toral subgroup of GL(V) is a subgroup in which can be simultaneously diagonalized, possibly requiring a field extension. A maximal torus is on which is not properly contained in any other.

3 The Singer Cycles S

The last point illustrate where the remaining pieces of the order of GL(V) come from. When we accounted for (q-1)d we used diagonal matrices. However we could have used any subgroup of matrices which can be diagonalized, and then choose a maximal such subgroup (proof that this is sufficient is beyond the scope at the momenet.) So we should ask, when is a matrix diagonalizable. The answer is when all the eigenvalues of the matrix lie in the field and the matrix is symmetricPlanetmathPlanetmathPlanetmathPlanetmath, meaning X=Xt.

As our fields GF(q) are finite, they are not algebraically closedMathworldPlanetmath, and so some matrices may have eigenvalues that are not in the field. For example:


has characteristic polynomialMathworldPlanetmathPlanetmath x2+1 which requires a -1. This is not an element of p when p is odd. Therefore we must create a quadratic extension to GF(p2) to diagonalize this matrix. In particular we would then be abel to write:


as the diagonalization of A. If p=2 the example


has irreduciblePlanetmathPlanetmath characteristic polynomial x2+x+1 and requires a quadratic fieldMathworldPlanetmath extensionPlanetmathPlanetmathPlanetmath as well. Notice that |B|=3 just as a generatorPlanetmathPlanetmathPlanetmath of GF(4)× requires.

When one extends to a field of order p2 then the multiplicative subgroup of GF(p2)× has order p2-1=(p-1)(p+1). Here we finally see the source of the remaining parts of the order of GL(V). We now collect this idea into a theorem.

Proposition 7.

GL(d/e,qe) embeds in GL(d,q) for every e|d. In particular, GF(qe)× embeds in GL(d,q) for every e|d.


Given W a d/e-dimensional vector spaceMathworldPlanetmath over GF(qe) where e|d, then W is a d-dimensional vector space over GF(q) as GF(q)GF(qe). Furthermore, if f:WW is a GF(qe)-linear map, then it is also a GF(q)-linear map. Thus we can interpret GL(d/e,qe) as a subgroup of GL(d,q).

To see that GF(qe)× embeds in GL(d,q) simply observe that the scalar matrices of GL(d/e,qe) are isomorphic to GF(qe). ∎

Caution: although the scalar matrices of GL(e,qd/e) are central in GL(e,qd/e), then need not be central in GL(d,q).

Definition 8.

A Singer cycle is an element fGL(V) which is the image of a scalar transform of some field extension.

Remark 9.

It is not uncommon for Singer cycles to refer only to maximal orderMathworldPlanetmath scalar transforms.

Finally, as GL(i,q) embeds in GL(d,q) for every 1id we now conclude:

Corollary 10.

GF(qi) embeds in GL(d,q) for every 1id. Moreover, the order


are represented by the Singer cycles of GL(d,q).


GF(qi)× embeds in GL(d,q) and has order qi-1=(q-1)(qi1++q+1). As it is cyclic, it has only one q-1 order subgroup and this must therefore correspond to the GF(q)× subgroup. These are accounted for with toral subgroups we already mentioned. The remaining qi-1++q+1 cyclic subgroup of GF(qi)× are Singer cycles of GL(d,q). In particular, they account for the remaining portion of the order formula of GL(d,q). ∎

Unfortunately, encoding a Singer cycle as a matrix is typically difficult. One can expect this by considering that these matrices must all be matrices which are not triangular. Thus a geometric pattern is out of the question. Yet one will also observe that for this very reason, there are many Singer cycles so at random there is a high probablity of finding such an invertible matrix. This is a very useful tool in practical algorithms for matrices which rely on non-deterministic approaches to remain efficient.

If the matrices of GL(d,q) are interpreted as matrices of GL(d,q¯), where q¯ denotes the algebraic closure of the field GF(q), then every Singer cycle is diagonalizable as all the eigenvalues now lie in the field. In this case every matrix of GL(d,q) can be triangularized and so we only need to talk of unipotent and toral subgroups.

The study of Singer cycles has largely been replaced by the more generally applicable study of toral subgroups. Singer cycles generally do not have determinantMathworldPlanetmath 1 and so the study of these elements in projective and special groups is difficult. The study of semi-simple elements from Lie theory and algebraic groups replaces the use of Singer cycles.

Title theory from orders of classical groups
Canonical name TheoryFromOrdersOfClassicalGroups
Date of creation 2013-03-22 15:56:48
Last modified on 2013-03-22 15:56:48
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 10
Author Algeboy (12884)
Entry type DerivationPlanetmathPlanetmath
Classification msc 11E57