scholarly journals On the number of conjugacy classes of a primitive permutation group

2021 ◽  
pp. 1-22
Author(s):  
Daniele Garzoni ◽  
Nick Gill
Keyword(s):  
Permutation Group ◽  
Conjugacy Classes ◽  
Primitive Permutation Group

Let $G$ be a primitive permutation group of degree $n$ with nonabelian socle, and let $k(G)$ be the number of conjugacy classes of $G$ . We prove that either $k(G)< n/2$ and $k(G)=o(n)$ as $n\rightarrow \infty$ , or $G$ belongs to explicit families of examples.

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 number of conjugacy classes of a permutation group

2015 ◽  
Vol 133 ◽  
pp. 251-260 ◽  
Author(s):  
Martino Garonzi ◽  
Attila Maróti
Keyword(s):  
Permutation Group ◽  
Conjugacy Classes

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

An investigation of countable B-groups

1987 ◽  
Vol 102 (2) ◽  
pp. 223-231 ◽  
Author(s):  
P. Cameron ◽  
K. W. Johnson
Keyword(s):  
Permutation Group ◽  
Countable Group ◽  
General Question ◽  
Primitive Permutation Group ◽  
Square Root ◽  
Regular Subgroup

A group G is defined to be a B-group if any primitive permutation group which contains G as a regular subgroup is doubly transitive. In the case where G is finite the existence of families of B-groups has been established by Burnside, Schur, Wielandt and others and led to the investigation of S-rings. A survey of this work is given in [3], sections 13·7–13·12. In this paper the possibility of the existence of countable B-groups is discussed. Three distinct methods are given to embed a countable group as a regular subgroup of a simply primitive permutation group, and in each case a condition on the square root sets of elements of the group is necessary for the embedding to be carried out. It is easy to demonstrate that this condition is not sufficient, and the general question remains open.

SUBORBITS IN INFINITE PRIMITIVE PERMUTATION GROUPS

2001 ◽  
Vol 33 (5) ◽  
pp. 583-590 ◽  
Author(s):  
DAVID M. EVANS
Keyword(s):  
Permutation Group ◽  
Permutation Groups ◽  
Primitive Permutation Group ◽  
Infinite Cardinal

For every infinite cardinal κ, we construct a primitive permutation group which has a finite suborbit paired with a suborbit of size κ. This answers a question of Peter M. Neumann.

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