14,495 research outputs found
On conjugacy classes of subgroups of the general linear group and cyclic orbit codes
Orbit codes are a family of codes employable for communications on a random
linear network coding channel. The paper focuses on the classification of these
codes. We start by classifying the conjugacy classes of cyclic subgroups of the
general linear group. As a result, we are able to focus the study of cyclic
orbit codes to a restricted family of them.Comment: 5 pages; Submitted to IEEE International Symposium on Information
Theory (ISIT) 201
Conjugacy classes in maximal parabolic subgroups of general linear groups
We compute conjugacy classes in maximal parabolic subgroups of the general
linear group. This computation proceeds by reducing to a ``matrix problem''.
Such problems involve finding normal forms for matrices under a specified set
of row and column operations. We solve the relevant matrix problem in small
dimensional cases. This gives us all conjugacy classes in maximal parabolic
subgroups over a perfect field when one of the two blocks has dimension less
than 6. In particular, this includes every maximal parabolic subgroup of
GL_n(k) for n < 12 and k a perfect field. If our field is finite of size q, we
also show that the number of conjugacy classes, and so the number of
characters, of these groups is a polynomial in with integral coefficients.Comment: 23 pages, 6 figures. See also
http://zaphod.uchicago.edu/~murray/research/index.html . Submitted to Journal
of Algebr
Constructive homomorphisms for classical groups
Let Omega be a quasisimple classical group in its natural representation over
a finite vector space V, and let Delta be its normaliser in the general linear
group. We construct the projection from Delta to Delta/Omega and provide fast,
polynomial-time algorithms for computing the image of an element. Given a
discrete logarithm oracle, we also represent Delta/Omega as a group with at
most 3 generators and 6 relations. We then compute canonical representatives
for the cosets of Omega. A key ingredient of our algorithms is a new,
asymptotically fast method for constructing isometries between spaces with
forms. Our results are useful for the matrix group recognition project, can be
used to solve element conjugacy problems, and can improve algorithms to
construct maximal subgroups
Finite abelian subgroups of the Cremona group of the plane
This work presents the conjugacy classes of finite abelian subgroups of the
Cremona group of the plane. Using a well-known theory, this problem amounts to
the study of automorphism groups of some Del Pezzo surfaces and conic bundles.
We have thus to enumerate all the cases, which gives a long description, and
then to show whether two cases are distinct or not, using some conjugacy
invariants. For example, we use the non-rational curves fixed by one element of
the group, and the action of the whole group on these curves. From this
classification, we deduce a sequence of more general results on birational
transformations, as for example the existence of infinitely many conjugacy
classes of elements of order n, for any even number n, a result false in the
odd case. We prove also that a root of some linear transformation of finite
order is itself conjugate to a linear transformation.Comment: PHD Thesis, 189 pages, 34 figures, original text may be found at
http://www.unige.ch/cyberdocuments/theses2006/BlancJ/meta.htm
On properties of translation groups in the affine general linear group with applications to cryptography
The affine general linear group acting on a vector space over a prime field is a well-understood mathematical object. Its elementary abelian regular subgroups have recently drawn attention in applied mathematics thanks to their use in cryptography as a way to hide or detect weaknesses inside block ciphers. This paper is focused on building a convenient representation of their elements which suits better the purposes of the cryptanalyst. Several combinatorial counting formulas and a classification of their conjugacy classes are given as well.publishedVersio
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