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.

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).

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.

On transitive permutation groups with a subgroup satisfying a certain conjugacy condition

1984 ◽  
Vol 36 (1) ◽  
pp. 69-86 ◽  
Author(s):  
Cheryl E. Praeger
Keyword(s):  
Permutation Group ◽  
Permutation Groups ◽  
Transitive Permutation Group ◽  
Pronormal Subgroup ◽  
Natural Representation ◽  
Conjugacy Condition ◽  
Sylow Subgroups ◽  
Point Stabilizer

AbstractLet G be transitive permutation group of degree n and let K be a nontrivial pronormal subgroup of G (that is, for all g in G, K and Kg are conjugate in (K, Kg)). It is shown that K can fix at most ½(n – 1) points. Moreover if K fixes exactly ½(n – 1) points then G is either An or Sn, or GL(d, 2) in its natural representation where n = 2d-1 ≥ 7. Connections with a result of Michael O'Nan are dicussed, and an application to the Sylow subgroups of a one point stabilizer is given.

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 On Multiple Transitivity of Permutation Groups

1961 ◽  
Vol 18 ◽  
pp. 93-109 ◽  
Author(s):  
Tosiro Tsuzuku
Keyword(s):  
Symmetric Group ◽  
Permutation Group ◽  
Irreducible Character ◽  
Symmetric Groups ◽  
Transitive Group ◽  
Permutation Groups ◽  
Irreducible Characters ◽  
Interesting Theorem

It is well known that a doubly transitive group has an irreducible character χ1 such that χ1(R) = α(R) − 1 for any element R of and a quadruply transitive group has irreducible characters χ3 and χ3 such that χ2(R) = where α(R) and β(R) are respectively the numbers of one cycles and two cycles contained in R. G. Frobenius was led to this fact in the connection with characters of the symmetric groups and he proved the following interesting theorem: if a permutation group of degree n is t-ply transitive, then any irreducible character of the symmetric group of degree n with dimension at most equal to is an irreducible character of .

scholarly journals Sylow subgroups of transitive permutation groups II

1977 ◽  
Vol 23 (3) ◽  
pp. 329-332 ◽  
Author(s):  
Cheryl E. Praeger
Keyword(s):  
Permutation Group ◽  
Minimal Length ◽  
Permutation Groups ◽  
Transitive Permutation Group ◽  
Sylow Subgroups ◽  
Finite Set

AbstractLet G be a transitive permutation group on a finite set of n points, and let P be a Sylow p-subgroup of G for some prime p dividing |G|. We are concerned with finding a bound for the number f of points of the set fixed by P. Of all the orbits of P of length greater than one, suppose that the ones of minimal length have length q, and suppose that there are k orbits of P of length q. We show that f ≦ kp − ip(n), where ip(n) is the integer satisfying 1 ≦ ip(n) ≦ p and n + ip(n) ≡ 0(mod p). This is a generalisation of a bound found by Marcel Herzog and the author, and this new bound is better whenever P has an orbit of length greater than the minimal length q.

scholarly journals Prime localizations of Burger–Mozes-type groups

2018 ◽  
Vol 21 (2) ◽  
pp. 229-240
Author(s):  
Stephan Tornier
Keyword(s):  
Large Class ◽  
Permutation Group ◽  
Permutation Groups ◽  
Finite Degree ◽  
Sylow Subgroups

AbstractThis article concerns Burger–Mozes universal groups acting on regular trees locally like a given permutation group of finite degree. We also consider locally isomorphic generalizations of the former due to Le Boudec and Lederle. For a large class of such permutation groups and primespwe determine their localp-Sylow subgroups as well as subgroups of theirp-localization, which is identified as a group of the same type in certain cases.

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).

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 Affine Primitive Groups and Semisymmetric Graphs

10.37236/2549 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Hua Han ◽  
Zaiping Lu
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Infinite Family ◽  
Permutation Groups ◽  
Primitive Groups ◽  
The Third ◽  
Semisymmetric Graphs

In this paper, we investigate semisymmetric graphs which arise from affine primitive permutation groups. We give a characterization of such graphs, and then construct an infinite family of semisymmetric graphs which contains the Gray graph as the third smallest member. Then, as a consequence, we obtain a factorization,of the complete bipartite graph $K_{p^{sp^t},p^{sp^t}}$ into connected semisymmetric graphs, where $p$ is an prime, $1\le t\le s$ with $s\ge2$ while $p=2$.

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