theory from orders of classical groups
One formula for the order of can be given as:
In Orders of Classical Groups (http://planetmath.org/OrdersAndStructureOfClassicalGroups) we see an accounting for this orders from an elementary linear algebra perspective and from there derive order of related classical groups. However, many of these formulas can be explicitly observed with the group theoretic structure of the various classical groups. We explore this presently.
1 The Unipotent matrices
Given a Sylow -subgroup of has order and is isomorphic to the group of lower-unitriangular matrices.
Observe for any , . Therefore as . Therefore is relatively prime to the
factor of the order of . Therefore the order of a Sylow -subgroup of is
A unipotent subgroup of 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
We will now account for the term in the order.
In general theory, diagonal matrices are replaced with the term toral subgroup. This is on account of the fact that when , is homotopic to the circle, . Therefore, as a topology, is , that is, the -dimensional torus.
3 The Singer Cycles
The last point illustrate where the remaining pieces of the order of come from. When we accounted for 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 symmetric, meaning .
As our fields are finite, they are not algebraically closed, and so some matrices may have eigenvalues that are not in the field. For example:
has characteristic polynomial which requires a . This is not an element of when is odd. Therefore we must create a quadratic extension to to diagonalize this matrix. In particular we would then be abel to write:
as the diagonalization of . If the example
embeds in for every . In particular, embeds in for every .
To see that embeds in simply observe that the scalar matrices of are isomorphic to . ∎
Caution: although the scalar matrices of are central in , then need not be central in .
A Singer cycle is an element which is the image of a scalar transform of some field extension.
It is not uncommon for Singer cycles to refer only to maximal order scalar transforms.
Finally, as embeds in for every we now conclude:
embeds in for every . Moreover, the order
are represented by the Singer cycles of .
embeds in and has order . As it is cyclic, it has only one order subgroup and this must therefore correspond to the subgroup. These are accounted for with toral subgroups we already mentioned. The remaining cyclic subgroup of are Singer cycles of . In particular, they account for the remaining portion of the order formula of . ∎
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 are interpreted as matrices of , where denotes the algebraic closure of the field , then every Singer cycle is diagonalizable as all the eigenvalues now lie in the field. In this case every matrix of 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 determinant 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|
|Date of creation||2013-03-22 15:56:48|
|Last modified on||2013-03-22 15:56:48|
|Last modified by||Algeboy (12884)|