On the point stabilizer in a primitive permutation group

1973 ◽  
Vol 133 (2) ◽  
pp. 137-168 ◽  
Author(s):  
Wolfgang Knapp
Keyword(s):  
Permutation Group ◽  
Primitive Permutation Group ◽  
Point Stabilizer

scholarly journals Primitive permutation groups with a doubly transitive subconstituent

1988 ◽  
Vol 45 (1) ◽  
pp. 66-77 ◽  
Author(s):  
Cheryl E. Praeger
Keyword(s):  
Normal Subgroup ◽  
Simple Group ◽  
Permutation Group ◽  
Minimal Normal Subgroup ◽  
Permutation Groups ◽  
Primitive Permutation Group ◽  
Diagonal Action ◽  
Product Action ◽  
Finite Set ◽  
Point Stabilizer

AbstractLet Gbe a primitive permutation group on a finite set Ω. We investigate the subconstitutents of G, that is the permutation groups induced by a point stabilizer on its orbits in Ω, in the cases where Ghas a diagonal action or a product action on Ω. In particular we show in these cases that no subconstituent is doubly transitive. Thus if G has a doubly transitive subconstituent we show that G has a unique minimal normal subgroup N and either N is a nonabelian simple group or N acts regularly on Ω: we investigate further the case where N is regular on Ω.

Some generalized characters associated to a transitive permutation group

2020 ◽  
Vol 23 (3) ◽  
pp. 393-397
Author(s):  
Wolfgang Knapp ◽  
Peter Schmid
Keyword(s):  
Positive Integer ◽  
Permutation Group ◽  
Frobenius Group ◽  
Sufficient Conditions ◽  
Necessary Condition ◽  
Transitive Permutation Group ◽  
Point Stabilizer ◽  
Necessary And Sufficient ◽  
Regular Character ◽  
Permutation Character

AbstractLet G be a finite transitive permutation group of degree n, with point stabilizer {H\neq 1} and permutation character π. For every positive integer t, we consider the generalized character {\psi_{t}=\rho_{G}-t(\pi-1_{G})}, where {\rho_{G}} is the regular character of G and {1_{G}} the 1-character. We give necessary and sufficient conditions on t (and G) which guarantee that {\psi_{t}} is a character of G. A necessary condition is that {t\leq\min\{n-1,\lvert H\rvert\}}, and it turns out that {\psi_{t}} is a character of G for {t=n-1} resp. {t=\lvert H\rvert} precisely when G is 2-transitive resp. a Frobenius group.

scholarly journals Bounds on finite quasiprimitive permutation groups

2001 ◽  
Vol 71 (2) ◽  
pp. 243-258 ◽  
Author(s):  
Cheryl E. Praeger ◽  
Aner Shalev
Keyword(s):  
Normal Subgroup ◽  
Permutation Group ◽  
Minimum Degree ◽  
Permutation Groups ◽  
Primitive Permutation Group ◽  
Base Size ◽  
Nontrivial Normal Subgroup

AbstractA permutation group is said to be quasiprimitive if every nontrivial normal subgroup is transitive. Every primitive permutation group is quasiprimitive, but the converse is not true. In this paper we start a project whose goal is to check which of the classical results on finite primitive permutation groups also holds for quasiprimitive ones (possibly with some modifications). The main topics addressed here are bounds on order, minimum degree and base size, as well as groups containing special p-elements. We also pose some problems for further research.

scholarly journals On finite G-locally primitive graphs and the Weiss conjecture

2004 ◽  
Vol 70 (3) ◽  
pp. 353-356 ◽  
Author(s):  
Mohammad A. Iranmanesh
Keyword(s):  
Permutation Group ◽  
Primitive Permutation Group ◽  
Vertex Transitive ◽  
Connected Vertex

A graph Γ is said to be a G-locally primitive graph, for G ≥ Aut Γ, if for every vertex, α, the stabiliser Gα induces a primitive permutation group on Γ (α) the set of vertices adjacent to α. In 1978 Richard Weiss conjectured that there exists a function f: ℕ →ℕ such that for any finite connected vertex-transitive G-locally primitive graph of valency d and a vertex α of the graph, |Gα| ≥ f(d). The purpose of this paper is to prove that, in the case Soc(G) = Sz(q), the conjecture is true.

scholarly journals The theorem of Marggraff on primitive permutation groups which contain a cycle

1976 ◽  
Vol 15 (1) ◽  
pp. 125-128 ◽  
Author(s):  
Richard Levingston ◽  
D.E. Taylor
Keyword(s):  
Permutation Group ◽  
Elementary Proof ◽  
Permutation Groups ◽  
Primitive Permutation Group

A short elementary proof is given of the theorem of Marggraff which states that a primitive permutation group which contains a cycle fixing k points is (k+1)-fold transitive. It is then shown that the method of proof actually yields a generalization of Marggraff's theorem.

scholarly journals On the normal structure of a one-point stabilizer in a doubly transitive permutation group

1978 ◽  
Vol 25 (2) ◽  
pp. 145-166
Author(s):  
M. D. Atkinson ◽  
Cheryl E. Praeger
Keyword(s):  
Normal Subgroup ◽  
Permutation Group ◽  
Closed Subgroup ◽  
Normal Structure ◽  
Subject Classification ◽  
Transitive Permutation Group ◽  
Normal Extension ◽  
Extra Information ◽  
Finite Set ◽  
Point Stabilizer

Let G be a doubly transitive permutation group on a finite set Ω, and let Kα be a normal subgroup of the stabilizer Gα of a point α in Ω. If the action of Gα on the set of orbits of Kα in Ω − {α} is 2-primitive with kernel Kα it is shown that either G is a normal extension of PSL(3, q) or Kα ∩ Gγ is a strongly closed subgroup of Gαγ in Gα, where γ ∈ Ω − {α}. If in addition the action of Gα on the set of orbits of Kα is assumed to be 3-transitive, extra information is obtained using permutation theoretic and centralizer ring methods. In the case where Kα has three orbits in Ω − {α} strong restrictions are obtained on either the structure of G or the degrees of certain irreducible characters of G. Subject classification (Amer. Math. Soc. (MOS) 1970: 20 B 20, 20 B 25.

scholarly journals On the Minimal Degree of a Primitive Permutation Group

1998 ◽  
Vol 207 (1) ◽  
pp. 127-145 ◽  
Author(s):  
Robert Guralnick ◽  
Kay Magaard
Keyword(s):  
Permutation Group ◽  
Minimal Degree ◽  
Primitive Permutation Group
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