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

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.

Singer Groups

1970 ◽  
Vol 22 (3) ◽  
pp. 492-513 ◽  
Author(s):  
Marshall D. Hestenes
Keyword(s):  
Projective Space ◽  
Systematic Study ◽  
Geometric Property ◽  
Difference Sets ◽  
Permutation Groups ◽  
Linear Groups ◽  
Singer Group ◽  
Singer Groups

Interest in the Singer groups has arisen in various places. The name itself results from the connection Singer [7] made between these groups and perfect difference sets, and this is closely associated with the geometric property that a Singer group is regular on the points of a projective space. Some information about these groups appears in Huppert's book [3, p. 187]. Singer groups are frequently useful in constructing examples and counterexamples. Our aim in this paper is to make a systematic study of the Singer subgroups of the linear groups, with a particular view to analyzing the examples they provide of Frobenius regular groups. Frobenius regular groups are a class of permutation groups generalizing the Zassenhaus groups, and Keller [5] has shown recently that they provide a new characterization of A6 and M11.

scholarly journals Coprime subdegrees for primitive permutation groups and completely reducible linear groups

2012 ◽  
Vol 195 (2) ◽  
pp. 745-772 ◽  
Author(s):  
Silvio Dolfi ◽  
Robert Guralnick ◽  
Cheryl E. Praeger ◽  
Pablo Spiga
Keyword(s):  
Permutation Groups ◽  
Linear Groups ◽  
Completely Reducible

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.

scholarly journals SOME INFINITE PERMUTATION GROUPS AND RELATED FINITE LINEAR GROUPS

2016 ◽  
Vol 102 (1) ◽  
pp. 136-149 ◽  
Author(s):  
PETER M. NEUMANN ◽  
CHERYL E. PRAEGER ◽  
SIMON M. SMITH
Keyword(s):  
Normal Subgroup ◽  
Vector Space ◽  
Finite Rank ◽  
Minimal Condition ◽  
Permutation Groups ◽  
Linear Groups ◽  
Normal Subgroups ◽  
Second Variation ◽  
Formula Group ◽  
Finite Linear

This article began as a study of the structure of infinite permutation groups $G$ in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point stabilisers are merely assumed to satisfy min-n, the minimal condition on normal subgroups. The groups $G$ are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal nontrivial normal subgroups, or they have a regular normal subgroup $M$ which is a divisible abelian $p$-group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on a $p$-adic vector space associated with $M$. This leads to our second variation, which is a study of the finite linear groups that can arise.

scholarly journals Bounding the composition length of primitive permutation groups and completely reducible linear groups

2018 ◽  
Vol 98 (3) ◽  
pp. 557-572
Author(s):  
S. P. Glasby ◽  
Cheryl E. Praeger ◽  
Kyle Rosa ◽  
Gabriel Verret
Keyword(s):  
Permutation Groups ◽  
Linear Groups ◽  
Completely Reducible

scholarly journals Asymptotic Results for Primitive Permutation Groups and Irreducible Linear Groups

2000 ◽  
Vol 223 (1) ◽  
pp. 154-170 ◽  
Author(s):  
A. Lucchini ◽  
F. Menegazzo ◽  
M. Morigi
Keyword(s):  
Permutation Groups ◽  
Linear Groups ◽  
Asymptotic Results
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