In the light of the presentations exist in the literature for the general and special linear group GLn(q) , SLn(q) [based on the construction given by Chevalley and Steinberg as well as on the concept of BN-pair syetem] such as http://dl.dropbox.com/u/5188175/slnpresentations.pdf , is there a convenient way for listing the elements of the general (or special) linear group GLn(q) over finite field in general or at least for GL2(q) or SL2(q) ?
please check the following link:
http://www.hull.ac.uk/php/masrs/2003/clgln2e2.pdf
Dear Ali, please if you can give me a reference on listing elements of the Dihedral group.
dear ali my idea is that the classification is no possible by presentations..
Dear Cenap,
What I am seeking is a convenient listing for the elements of those two groups, I am not considering their representations. Of course such listing could help in constructing the permutation representation for those groups though this is not the aim.
Dear Hanifa,
The listing of the elements for the Dihedral group can be found in any standard text book in group theory; for example : J. S Rose, A course On Group Theory p.27
Dear Cenap
Thanks for the reference
http://www.hull.ac.uk/php/masrs/2003/clgln2e2.pdf
I need to read it thoroughly though I could not spot any listing there (please correct me) only counting.
Dear Cenap,
Thanks for the other paper by M.R. Darafsheh. This paper only deals with the order of elements.
Dear Ali, thank you for the reference. I read page 26 and 27 about the Dihedral group and see the way it listed its elements: 1, ρ, ..., ρn-1 , ε, ρε,..., ρn-1ε, where ρ and ε are the two generators of D2n of order n and 2 resp.
So, we can list elements of GL(n,q) by taking the generators with their order and use their orders to find all "single" elements with their powers, then building the semi-direct products to have the the combinations. I think this is the way. Of course, it is not easy like the Dihedral group, it returns to the pi -Sylow groups and their generators, but with transvections and Dilations as generators of GL(n,q) with matrix calculation, may be it become "affordable"
Dear Hanifa
Thanks for your comment. In principle you are right but in practice the fact that the Dihedral has a "nice" presentation makes it easy to list the elements and I think one needs "as nice" presentation for GL(n,q).
Thanks AHMED Munir,
For G=GL(2, q) (being a group with BN-pair), it is possible to list the elements of G using its Bruhat decomposition as disjoint union of the (B,B)-double cosets in G (there are only two such double cosets corresponding to the symmetric group S2 ;the Coxeter group of the BN-pair system of G-), but for a n greater than 2 we may need different method.
how is the Groebner - Shirshov basis for the classification using diamond lemma
take a look on this notes
http://www.maths.qmul.ac.uk/~raw/FSG/notes2.pdf
Also this link used Bruhat decomposition for a Gln and Sln , but over an algebraically closed field
https://en.wikipedia.org/wiki/Bruhat_decomposition
THANKS TO YOU ALL ;
It turns out that the BN-pair structure of the general linear group G=GL(2,q)=B U BwB can be used to find elements listing for GL(2,q) appropriate for finding the RG-matrix for this group which is in turn used for finding the ring of |GL(2,q)xGL(2,q)| matrices representing the group ring of GL(2,q). The (zero-divisor and unit) elements of this ring of matrices are usually used to construct linear codes. This group GL(2,q) (as far as i know) is the first linear group to be considered for this purpose, the RG-matrix has a block shape but we do not know yet what kind of codes will be obtained as an outcome of this identification.
This paper
http://dergipark.gov.tr/download/article-file/603978
Contains a listing for the elements of the general linear group GL(2,q) which proves to be quite appropriate for finding the “coding matrix” of this group. This coding matrix is essential in identifying the group algebra kGL(2,q) with a ring of matrices (identification proved by T. Hurley), which in turn being used for constructing codes out of zero divisors and unit elements in the group algebra.
- Badges
- Science topic
- Similar topics
- Social Psychology
- Attitude