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 Transversals in permutation groups and factorisations of complete graphs

2002 ◽  
Vol 65 (2) ◽  
pp. 277-288 ◽  
Author(s):  
Gil Kaplan ◽  
Arieh Lev
Keyword(s):  
Permutation Group ◽  
Directed Graph ◽  
Sufficient Conditions ◽  
Combinatorial Problems ◽  
Permutation Groups ◽  
Transitive Permutation Group ◽  
Complete Graphs ◽  
Frobenius Theorem ◽  
Disjoint Cycles ◽  
Finite Set

Let G be a transitive permutation group acting on a finite set of order n. We discuss certain types of transversals for a point stabiliser A in G: free transversals and global transversals. We give sufficient conditions for the existence of such transversals, and show the connection between these transversals and combinatorial problems of decomposing the complete directed graph into edge disjoint cycles. In particular, we classify all the inner-transitive Oberwolfach factorisations of the complete directed graph. We mention also a connection to Frobenius theorem.

scholarly journals On Primitive Extensions of Rank 3 of Symmetric Groups

1966 ◽  
Vol 27 (1) ◽  
pp. 171-177 ◽  
Author(s):  
Tosiro Tsuzuku
Keyword(s):  
Permutation Group ◽  
Symmetric Groups ◽  
Transitive Group ◽  
Permutation Groups ◽  
Minimal Set ◽  
Finite Set ◽  
Rank 2 ◽  
One To One

1. Let Ω be a finite set of arbitrary elements and let (G, Ω) be a permutation group on Ω. (This is also simply denoted by G). Two permutation groups (G, Ω) and (G, Γ) are called isomorphic if there exist an isomorphism σ of G onto H and a one to one mapping τ of Ω onto Γ such that (g(i))τ=gσ(iτ) for g ∊ G and i∊Ω. For a subset Δ of Ω, those elements of G which leave each point of Δ individually fixed form a subgroup GΔ of G which is called a stabilizer of Δ. A subset Γ of Ω is called an orbit of GΔ if Γ is a minimal set on which each element of G induces a permutation. A permutation group (G, Ω) is called a group of rank n if G is transitive on Ω and the number of orbits of a stabilizer Ga of a ∊ Ω, is n. A group of rank 2 is nothing but a doubly transitive group and there exist a few results on structure of groups of rank 3 (cf. H. Wielandt [6], D. G. Higman M).

S-rings over loops, right mapping groups and transversals in permutation groups

1981 ◽  
Vol 89 (3) ◽  
pp. 433-443 ◽  
Author(s):  
K. W. Johnson
Keyword(s):  
Permutation Group ◽  
Group Ring ◽  
Permutation Representation ◽  
Permutation Groups ◽  
P 75 ◽  
Regular Subgroup ◽  
Finite Set

The centralizer ring of a permutation representation of a group appears in several contexts. In (19) and (20) Schur considered the situation where a permutation group G acting on a finite set Ω has a regular subgroup H. In this case Ω may be given the structure of H and the centralizer ring is isomorphic to a subring of the group ring of H. Schur used this in his investigations of B-groups. A group H is a B-group if whenever a permutation group G contains H as a regular subgroup then G is either imprimitive or doubly transitive. Surveys of the results known on B-groups are given in (28), ch. IV and (21), ch. 13. In (28), p. 75, remark F, it is noted that the existence of a regular subgroup is not necessary for many of the arguments. This paper may be regarded as an extension of this remark, but the approach here differs slightly from that suggested by Wielandt in that it appears to be more natural to work with transversals rather than cosets.

scholarly journals Finding common fixed points of nonexpansive mappings by iteration

2000 ◽  
Vol 62 (2) ◽  
pp. 307-310 ◽  
Author(s):  
B. E. Rhoades
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Common Fixed Point ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Iteration Scheme ◽  
Finite Set

In this paper it is shown that a particular iteration scheme converges weakly to a common fixed point of a finite set of nonexpansive mappings. This result is an improvement of two related theorems in the literature.

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 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 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.)

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 Solvability of a Class of Rank 3 Permutation Groups

1971 ◽  
Vol 41 ◽  
pp. 89-96 ◽  
Author(s):  
D.G. Higman
Keyword(s):  
Automorphism Group ◽  
Permutation Group ◽  
Regular Graph ◽  
Strongly Regular Graph ◽  
Permutation Groups ◽  
Theoretical Interpretation ◽  
Finite Set ◽  
Strongly Regular ◽  
Even Order

1. Introduction. Let G be a rank 3 permutation group of even order on a finite set X, |X| = n, and let Δ and Γ be the two nontrivial orbits of G in X×X under componentwise action. As pointed out by Sims [6], results in [2] can be interpreted as implying that the graph = (X, Δ) is a strongly regular graph, the graph theoretical interpretation of the parameters k, l, λ and μ of [2] being as follows: k is the degree of , λ is the number of triangles containing a given edge, and μ is the number of paths of length 2 joining a given vertex P to each of the l vertices ≠ P which are not adjacent to P. The group G acts as an automorphism group on and on its complement = (X,Γ).

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