scholarly journals The Classification of Permutation Groups with Maximum Orbits

2018 ◽  
Vol 15 ◽  
pp. 8155-8161
Author(s):  
Behname Razzaghmaneshi
Keyword(s):  
Positive Integer ◽  
Fixed Points ◽  
Permutation Group ◽  
Permutation Groups

Let G be a permutation group on a set with no fixed points in and let m be a positive integer. If no element of G moves any subset of by more than m points (that is, if for every and g 2 G), and the lengths two of orbits is p, and the restof orbits have lengths equal to 3. Then the number t of G-orbits in is at most  Moreover, we classifiy all groups for is hold.(For  denotes the greatest integer less than or equal to x.)

scholarly journals Intransitive Permutation Groups with Bounded Movement Having Maximum Degree

2019 ◽  
Vol 16 ◽  
pp. 8272-8279
Author(s):  
Behnam Razzagh
Keyword(s):  
Positive Integer ◽  
Fixed Points ◽  
Permutation Group ◽  
Maximum Degree ◽  
Permutation Groups ◽  
Ams Classification ◽  
Bounded Movement

Let G be a permutation group on a set with no fixed points in and let m be a positive integer. If for each subset of  the size  is bounded, for , we define the movement of g as the max  over all subsets of . In this paper we classified all of permutation groups on set of size 3m + 1 with 2 orbits such that has movement m . 2000 AMS classification subjects: 20B25

scholarly journals Improvement on the bounds of permutation groups with bounded movement

2003 ◽  
Vol 67 (2) ◽  
pp. 249-256 ◽  
Author(s):  
Mehdi Alaeiyan
Keyword(s):  
Positive Integer ◽  
Fixed Points ◽  
Permutation Group ◽  
Infinite Family ◽  
Permutation Groups ◽  
Bounded Movement

Let G be a permutation group on a set Ω with no fixed points in Ω and let m be a positive integer. Then we define the movement of G as, m := move(G) := supΓ{|Γg \ Γ| │ g ∈ G}. Let p be a prime, p ≥ 5. If G is not a 2-group and p is the least odd prime dividing |G|, then we show that n := |Ω| ≤ 4m – p + 3.Moreover, if we suppose that the permutation group induced by G on each orbit is not a 2-group then we improve the last bound of n and for an infinite family of groups the bound is attained.

scholarly journals Intransitive Permutation Groups with Bounded Movement Having Maximum Degree

2019 ◽  
Vol 16 ◽  
pp. 8340-8347
Author(s):  
Behnam Razzagh
Keyword(s):  
Positive Integer ◽  
Fixed Points ◽  
Permutation Group ◽  
Maximum Degree ◽  
Permutation Groups ◽  
Ams Classification ◽  
Bounded Movement

Let G be a permutation group on a set withno fixed points in and let m be a positive integer. If for each subset T of the  size |Tg\T| is bounded, for gEG, we define the movement of g as the max|Tg\T| over all subsets T of . In this paper we classified all of permutation groups on set    of size 3m + 1 with 2 orbits such that has movement m . 2000 AMS classification subjects: 20B25

MULTIPLE TRANSITIVITY AND MIN-WISE INDEPENDENCE IN PERMUTATION GROUPS

2004 ◽  
Vol 03 (04) ◽  
pp. 427-435
Author(s):  
C. FRANCHI
Keyword(s):  
Positive Integer ◽  
Permutation Group ◽  
Ordered Set ◽  
Permutation Groups ◽  
Linearly Ordered Set

Let Ω be a finite linearly ordered set and let k be a positive integer. A permutation group G on Ω is called co-k-restricted min-wise independent on Ω if [Formula: see text] for any X⊆Ω such that |X|≥|Ω|-k+1 and for any x∈X. We show that co-k-restricted min-wise independent groups are exactly the groups with the property that for each subset X⊆Ω with |X|≤k-1, the stabilizer G{X} of X in G is transitive on Ω\X. Using this fact, we determine all co-k-restricted min-wise independent 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.

scholarly journals On the minimal degree of a transitive permutation group with stabilizer a 2-group

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Primož Potočnik ◽  
Pablo Spiga
Keyword(s):  
Fixed Points ◽  
Permutation Group ◽  
Minimal Degree ◽  
Simple Groups ◽  
Transitive Permutation Group ◽  
Finite Simple Groups ◽  
Trivial Element ◽  
Stabilizer Of A Point

AbstractThe minimal degree of a permutation group 𝐺 is defined as the minimal number of non-fixed points of a non-trivial element of 𝐺. In this paper, we show that if 𝐺 is a transitive permutation group of degree 𝑛 having no non-trivial normal 2-subgroups such that the stabilizer of a point is a 2-group, then the minimal degree of 𝐺 is at least \frac{2}{3}n. The proof depends on the classification of finite simple groups.

scholarly journals Statistics on the Multi-Colored Permutation Groups

10.37236/942 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Eli Bagno ◽  
Ayelet Butman ◽  
David Garber
Keyword(s):  
Fixed Points ◽  
Permutation Group ◽  
Wreath Product ◽  
Fixed Number ◽  
Permutation Groups

We define an excedance number for the multi-colored permutation group i.e. the wreath product $({\Bbb Z}_{r_1} \times \cdots \times {\Bbb Z}_{r_k}) \wr S_n$ and calculate its multi-distribution with some natural parameters. We also compute the multi–distribution of the parameters exc$(\pi)$ and fix$(\pi)$ over the sets of involutions in the multi-colored permutation group. Using this, we count the number of involutions in this group having a fixed number of excedances and absolute fixed points.

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.

On the fixed points of Sylow subgroups of transitive permutation groups

1976 ◽  
Vol 21 (4) ◽  
pp. 428-437 ◽  
Author(s):  
Marcel Herzog ◽  
Cheryl E. Praeger
Keyword(s):  
Fixed Points ◽  
Permutation Group ◽  
Upper Bound ◽  
Permutation Groups ◽  
Transitive Permutation Group ◽  
Sylow Subgroups

AbstractLet G be a transitive permutation group on a set Ω of n points, and let P be a Sylow p-subgroup of G for some prime p dividing ∣G∣. If P has t long orbits and f fixed points in Ω, then it is shown that f ≦ tp − ip(n), where ip(n) = p – rp(n), rp(n) denoting the residue of n modulo p. In addition, groups for which f attains the upper bound are classified.

scholarly journals Coprime subdegrees of twisted wreath permutation groups

2019 ◽  
Vol 62 (4) ◽  
pp. 1137-1162
Author(s):  
Alexander Y. Chua ◽  
Michael Giudici ◽  
Luke Morgan
Keyword(s):  
Simple Group ◽  
Permutation Group ◽  
Permutation Groups ◽  
Primitive Permutation Group ◽  
Primitive Groups ◽  
Group A ◽  
Full Classification

AbstractDolfi, Guralnick, Praeger and Spiga asked whether there exist infinitely many primitive groups of twisted wreath type with non-trivial coprime subdegrees. Here, we settle this question in the affirmative. We construct infinite families of primitive twisted wreath permutation groups with non-trivial coprime subdegrees. In particular, we define a primitive twisted wreath group G(m, q) constructed from the non-abelian simple group PSL(2, q) and a primitive permutation group of diagonal type with socle PSL(2, q)m, and determine many subdegrees for this group. A consequence is that we determine all values of m and q for which G(m, q) has non-trivial coprime subdegrees. In the case where m = 2 and $q\notin \{7,11,29\}$, we obtain a full classification of all pairs of non-trivial coprime subdegrees.

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