scholarly journals The classification of $\frac {3}{2}$-transitive permutation groups and $\frac {1}{2}$-transitive linear groups

2019 ◽  
Vol 147 (12) ◽  
pp. 5023-5037 ◽  
Author(s):  
Martin W. Liebeck ◽  
Cheryl E. Praeger ◽  
Jan Saxl
Keyword(s):  
Permutation Groups ◽  
Linear Groups ◽  
Transitive Linear

scholarly journals Some Constant Weight Codes from Primitive Permutation Groups

10.37236/2702 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Derek H. Smith ◽  
Roberto Montemanni
Keyword(s):  
Automorphism Group ◽  
Fixed Points ◽  
Maximum Clique ◽  
Permutation Groups ◽  
Linear Groups ◽  
Web Pages ◽  
Constant Weight Codes ◽  
Constant Weight ◽  
General Linear Groups

In recent years the detailed study of the construction of constant weight codes has been extended from length at most 28 to lengths less than 64. Andries Brouwer maintains web pages with tables of the best known constant weight codes of these lengths. In many cases the codes have more codewords than the best code in the literature, and are not particularly easy to improve. Many of the codes are constructed using a specified permutation group as automorphism group. The groups used include cyclic, quasi-cyclic, affine general linear groups etc. sometimes with fixed points. The precise rationale for the choice of groups is not clear.In this paper the choice of groups is made systematic by the use of the classification of primitive permutation groups. Together with several improved techniques for finding a maximum clique, this has led to the construction of 39 improved constant weight codes.

Primitive Genus Two Groups of Meta Type

2018 ◽  
Vol 7 (1-2) ◽  
pp. 1
Author(s):  
Haval Mohammed Salih
Keyword(s):  
Computer Algebra ◽  
Computer Algebra System ◽  
Complete Classification ◽  
Permutation Groups ◽  
Affine Groups ◽  
Two Systems ◽  
Genus Two

This paper is a contribution to the classification of the finite primitive permutation groups of genus two. We consider the case of affine groups. Our main result, Lemma 3.10 gives a complete classification of genus two systems when . We achieve this classification with the aid of the computer algebra system GAP.

TRANSITIVE PERMUTATION GROUPS AND IRREDUCIBLE LINEAR GROUPS

1995 ◽  
Vol 46 (4) ◽  
pp. 385-407 ◽  
Author(s):  
R. M. BRYANT ◽  
L. G. KOVÁCS ◽  
G. R. ROBINSON
Keyword(s):  
Permutation Groups ◽  
Linear Groups

Classification of Infinite Primitive Jordan Permutation Groups

1996 ◽  
Vol s3-72 (1) ◽  
pp. 63-123 ◽  
Author(s):  
S. A. Adeleke ◽  
Dugald Macpherson
Keyword(s):  
Permutation Groups

scholarly journals THE -GENERATION OF THE SPECIAL LINEAR GROUPS OVER FINITE FIELDS

2016 ◽  
Vol 95 (1) ◽  
pp. 48-53 ◽  
Author(s):  
MARCO ANTONIO PELLEGRINI
Keyword(s):  
Finite Fields ◽  
Simple Groups ◽  
Linear Groups ◽  
Finite Simple Groups ◽  
Special Linear ◽  
Special Linear Groups

We complete the classification of the finite special linear groups $\text{SL}_{n}(q)$ which are $(2,3)$-generated, that is, which are generated by an involution and an element of order $3$. This also gives the classification of the finite simple groups $\text{PSL}_{n}(q)$ which are $(2,3)$-generated.

The Permutation Group Theory and Electrons Interaction

2003 ◽  
Vol 17 (21) ◽  
pp. 3813-3830 ◽  
Author(s):  
K. D. Tovstyuk ◽  
C. C. Tovstyuk ◽  
O. O. Danylevych
Keyword(s):  
Green's Functions ◽  
Green’S Functions ◽  
Effective Interaction ◽  
Momentum Conservation ◽  
Permutation Groups ◽  
Interacting Electrons ◽  
Virtual Interaction ◽  
Initial Interaction ◽  
Energy And Momentum

The new mathematical formalism for the Green's functions of interacting electrons in crystals is constructed. It is based on the theory of Green's functions and permutation groups. We constructed a new object of permutation groups, which we call double permutation (DP). DP allows one to take into consideration the symmetry of the ground state as well as energy and momentum conservation in every virtual interaction. We developed the classification of double permutations and proved the theorem, which allows the selection of classes of associated double permutations (ADP). The Green's functions are constructed for series of ADP. We separate in the DP the convolving columns by replacing the initial interaction between the particles with the effective interaction. In convoluting the series for Green's functions, we use the methods developed for permutation groups schemes of Young–Yamanuti.

scholarly journals Invariant Relations and Aschbacher Classes of Finite Linear Groups

10.37236/712 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Jing Xu ◽  
Michael Giudici ◽  
Cai Heng Li ◽  
Cheryl E. Praeger
Keyword(s):  
Positive Integer ◽  
Vector Space ◽  
Permutation Group ◽  
Cartesian Product ◽  
Permutation Groups ◽  
Linear Groups ◽  
Natural Action ◽  
Fourth Author ◽  
Empty Subset ◽  
Finite Linear

For a positive integer $k$, a $k$-relation on a set $\Omega$ is a non-empty subset $\Delta$ of the $k$-fold Cartesian product $\Omega^k$; $\Delta$ is called a $k$-relation for a permutation group $H$ on $\Omega$ if $H$ leaves $\Delta$ invariant setwise. The $k$-closure $H^{(k)}$ of $H$, in the sense of Wielandt, is the largest permutation group $K$ on $\Omega$ such that the set of $k$-relations for $K$ is equal to the set of $k$-relations for $H$. We study $k$-relations for finite semi-linear groups $H\leq{\rm\Gamma L}(d,q)$ in their natural action on the set $\Omega$ of non-zero vectors of the underlying vector space. In particular, for each Aschbacher class ${\mathcal C}$ of geometric subgroups of ${\rm\Gamma L}(d,q)$, we define a subset ${\rm Rel}({\mathcal C})$ of $k$-relations (with $k=1$ or $k=2$) and prove (i) that $H$ lies in ${\mathcal C}$ if and only if $H$ leaves invariant at least one relation in ${\rm Rel}({\mathcal C})$, and (ii) that, if $H$ is maximal among subgroups in ${\mathcal C}$, then an element $g\in{\rm\Gamma L}(d,q)$ lies in the $k$-closure of $H$ if and only if $g$ leaves invariant a single $H$-invariant $k$-relation in ${\rm Rel}({\mathcal C})$ (rather than checking that $g$ leaves invariant all $H$-invariant $k$-relations). Consequently both, or neither, of $H$ and $H^{(k)}\cap{\rm\Gamma L}(d,q)$ lie in ${\mathcal C}$. As an application, we improve a 1992 result of Saxl and the fourth author concerning closures of affine primitive permutation groups.

scholarly journals A Note on Primitive Permutation Groups of Prime Power Degree

2015 ◽  
Vol 2015 ◽  
pp. 1-4
Author(s):  
Qian Cai ◽  
Hua Zhang
Keyword(s):  
Prime Power ◽  
Permutation Groups ◽  
Simple Type ◽  
Present Stage ◽  
Primitive Groups ◽  
Short Survey ◽  
Product Action ◽  
Action Type ◽  
Affine Type

Primitive permutation groups of prime power degree are known to be affine type, almost simple type, and product action type. At the present stage finding an explicit classification of primitive groups of affine type seems untractable, while the product action type can usually be reduced to almost simple type. In this paper, we present a short survey of the development of primitive groups of prime power degree, together with a brief description on such groups.

scholarly journals Unitary dual of GL(n) at archimedean places and global Jacquet–Langlands correspondence

2010 ◽  
Vol 146 (5) ◽  
pp. 1115-1164 ◽  
Author(s):  
A. I. Badulescu ◽  
D. Renard
Keyword(s):  
General Linear ◽  
Linear Groups ◽  
Automorphic Representations ◽  
Langlands Correspondence ◽  
General Linear Groups ◽  
Multiplicity One ◽  
Unitary Dual ◽  
Strong Multiplicity ◽  
Local Results

AbstractIn a paper by Badulescu [Global Jacquet–Langlands correspondence, multiplicity one and classification of automorphic representations, Invent. Math. 172 (2008), 383–438], results on the global Jacquet–Langlands correspondence, (weak and strong) multiplicity-one theorems and the classification of automorphic representations for inner forms of the general linear group over a number field were established, under the assumption that the local inner forms are split at archimedean places. In this paper, we extend the main local results of that article to archimedean places so that the above condition can be removed. Along the way, we collect several results about the unitary dual of general linear groups over ℝ, ℂ or ℍ which are of independent interest.

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