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. 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 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

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.)

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 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.

Trivial Set-Stabilizers in Finite Permutation Groups

1983 ◽  
Vol 35 (1) ◽  
pp. 59-67 ◽  
Author(s):  
David Gluck
Keyword(s):  
Permutation Group ◽  
Symmetric Groups ◽  
Infinite Family ◽  
Permutation Groups ◽  
Alternating Groups ◽  
Sylow Subgroups ◽  
Odd Order

For which permutation groups does there exist a subset of the permuted set whose stabilizer in the group is trivial?The permuted set has so many subsets that one might expect that subsets with trivial stabilizer usually exist. The symmetric and alternating groups are obvious exceptions to this expectation. Another, more interesting, infinite family of exceptions are the 2-Sylow subgroups of the symmetric groups on 2n symbols, in their natural representations on 2n points.One of our main results, Corollary 1, sheds some light on this last family of groups. We show that when the permutation group has odd order, there is indeed a subset of the permuted set whose stabilizer in the group is trivial. Corollary 1 follows easily from Theorem 1, which completely classifies the primitive solvable permutation groups in which every subset of the permuted set has non-trivial stabilizer.

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 Fixed Point Polynomials of Permutation Groups

10.37236/2955 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
C. M. Harden ◽  
D. B. Penman
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Unit Circle ◽  
Permutation Group ◽  
Large Family ◽  
Permutation Groups ◽  
Future Directions ◽  
Real Roots ◽  
Finite Set ◽  
Root Location

In this paper we study, given a group $G$ of permutations of a finite set, the so-called fixed point polynomial $\sum_{i=0}^{n}f_{i}x^{i}$, where $f_{i}$ is the number of permutations in $G$ which have exactly $i$ fixed points. In particular, we investigate how root location relates to properties of the permutation group. We show that for a large family of such groups most roots are close to the unit circle and roughly uniformly distributed round it. We prove that many families of such polynomials have few real roots. We show that many of these polynomials are irreducible when the group acts transitively. We close by indicating some future directions of this research. A corrigendum was appended to this paper on 10th October 2014. 

scholarly journals ON THE HEIGHT AND RELATIONAL COMPLEXITY OF A FINITE PERMUTATION GROUP

2021 ◽  
pp. 1-40
Author(s):  
NICK GILL ◽  
BIANCA LODÀ ◽  
PABLO SPIGA
Keyword(s):  
Permutation Group ◽  
Model Theory ◽  
Independent Set ◽  
Maximum Size ◽  
Proper Subset ◽  
Permutation Groups ◽  
Finite Permutation Group ◽  
Pointwise Stabilizer ◽  
Relational Complexity

Abstract Let G be a permutation group on a set $\Omega $ of size t. We say that $\Lambda \subseteq \Omega $ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $\Lambda $ . We define the height of G to be the maximum size of an independent set, and we denote this quantity $\textrm{H}(G)$ . In this paper, we study $\textrm{H}(G)$ for the case when G is primitive. Our main result asserts that either $\textrm{H}(G)< 9\log t$ or else G is in a particular well-studied family (the primitive large–base groups). An immediate corollary of this result is a characterization of primitive permutation groups with large relational complexity, the latter quantity being a statistic introduced by Cherlin in his study of the model theory of permutation groups. We also study $\textrm{I}(G)$ , the maximum length of an irredundant base of G, in which case we prove that if G is primitive, then either $\textrm{I}(G)<7\log t$ or else, again, G is in a particular family (which includes the primitive large–base groups as well as some others).

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