scholarly journals On the Saxl graph of a permutation group

2018 ◽  
Vol 168 (2) ◽  
pp. 219-248 ◽  
Author(s):  
TIMOTHY C. BURNESS ◽  
MICHAEL GIUDICI
Keyword(s):  
Permutation Group ◽  
Large Diameter ◽  
Probabilistic Approach ◽  
Transitive Group ◽  
Primitive Group ◽  
Simple Groups ◽  
Connected Components ◽  
Open Problems ◽  
Almost Simple Groups ◽  
Independence Numbers

AbstractLet G be a permutation group on a set Ω. A subset of Ω is a base for G if its pointwise stabiliser in G is trivial. In this paper we introduce and study an associated graph Σ(G), which we call the Saxl graph of G. The vertices of Σ(G) are the points of Ω, and two vertices are adjacent if they form a base for G. This graph encodes some interesting properties of the permutation group. We investigate the connectivity of Σ(G) for a finite transitive group G, as well as its diameter, Hamiltonicity, clique and independence numbers, and we present several open problems. For instance, we conjecture that if G is a primitive group with a base of size 2, then the diameter of Σ(G) is at most 2. Using a probabilistic approach, we establish the conjecture for some families of almost simple groups. For example, the conjecture holds when G = Sn or An (with n > 12) and the point stabiliser of G is a primitive subgroup. In contrast, we can construct imprimitive groups whose Saxl graph is disconnected with arbitrarily many connected components, or connected with arbitrarily large diameter.

scholarly journals ON THE RATIONALITY OF ALMOST SIMPLE ALGEBRAIC GROUPS

1999 ◽  
Vol 10 (05) ◽  
pp. 643-665
Author(s):  
NGUYÊÑ QUÔĆ THǍŃG
Keyword(s):  
Dynkin Diagram ◽  
Algebraic Groups ◽  
Simple Groups ◽  
Connected Components ◽  
Weak Approximation ◽  
Stable Rationality ◽  
Almost Simple Groups ◽  
Anisotropic Kernel

We prove the stable rationality of almost simple adjoint algebraic groups, the connected components of the Dynkin diagram of anisotropic kernel of which contain at most two vertices. The (stable) rationality of many isotropic almost simple groups with small anisotropic kernel and some related results in weak approximation over arbitrary fields are discussed.

scholarly journals Hurwitz components of groups with socle PSL(3, q)

2021 ◽  
Vol 36 (1) ◽  
pp. 51-62
Author(s):  
H.M. Mohammed Salih
Keyword(s):  
Finite Group ◽  
Monodromy Group ◽  
Riemann Sphere ◽  
Complete List ◽  
Simple Groups ◽  
Connected Components ◽  
Branch Points ◽  
Almost Simple Groups ◽  
Hurwitz Space ◽  
A Finite Group

For a finite group G, the Hurwitz space Hinr,g(G) is the space of genus g covers of the Riemann sphere P1 with r branch points and the monodromy group G. In this paper, we give a complete list of some almost simple groups of Lie rank two. That is, we assume that G is a primitive almost simple groups of Lie rank two. Under this assumption we determine the braid orbits on the suitable Nielsen classes, which is equivalent to finding connected components in Hinr,g(G).

scholarly journals On valency problems of Saxl graphs

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiyong Chen ◽  
Hong Yi Huang
Keyword(s):  
Permutation Group ◽  
Prime Power ◽  
Frobenius Group ◽  
Transitive Group ◽  
Primitive Group ◽  
Vertex Set ◽  
Vertex Transitive ◽  
Sigma G ◽  
Cyclic Kernel ◽  
General Method

Abstract Let 𝐺 be a permutation group on a set Ω, and recall that a base for 𝐺 is a subset of Ω such that its pointwise stabiliser is trivial. In a recent paper, Burness and Giudici introduced the Saxl graph of 𝐺, denoted Σ ⁢ ( G ) \Sigma(G) , with vertex set Ω and two vertices adjacent if and only if they form a base for 𝐺. If 𝐺 is transitive, then Σ ⁢ ( G ) \Sigma(G) is vertex-transitive, and it is natural to consider its valency (which we refer to as the valency of 𝐺). In this paper, we present a general method for computing the valency of any finite transitive group, and we use it to calculate the exact valency of every primitive group with stabiliser a Frobenius group with cyclic kernel. As an application, we calculate the valency of every almost simple primitive group with an alternating socle and soluble stabiliser, and we use this to extend results of Burness and Giudici on almost simple primitive groups with prime-power or odd valency.

Non-commuting graphs of certain almost simple groups

2019 ◽  
Vol 12 (05) ◽  
pp. 1950081
Author(s):  
M. Jahandideh ◽  
R. Modabernia ◽  
S. Shokrolahi
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Simple Group ◽  
Simple Groups ◽  
Almost Simple Group ◽  
Almost Simple Groups ◽  
Commuting Graph ◽  
Vertex Set

Let [Formula: see text] be a non-abelian finite group and [Formula: see text] be the center of [Formula: see text]. The non-commuting graph, [Formula: see text], associated to [Formula: see text] is the graph whose vertex set is [Formula: see text] and two distinct vertices [Formula: see text] are adjacent if and only if [Formula: see text]. We conjecture that if [Formula: see text] is an almost simple group and [Formula: see text] is a non-abelian finite group such that [Formula: see text], then [Formula: see text]. Among other results, we prove that if [Formula: see text] is a certain almost simple group and [Formula: see text] is a non-abelian group with isomorphic non-commuting graphs, then [Formula: see text].

CHARACTERIZATION BY PRIME GRAPH OF PGL(2, pk) WHERE p AND k > 1 ARE ODD

2010 ◽  
Vol 20 (07) ◽  
pp. 847-873 ◽  
Author(s):  
Z. AKHLAGHI ◽  
B. KHOSRAVI ◽  
M. KHATAMI
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Prime Number ◽  
Prime Graph ◽  
Composition Factor ◽  
Simple Groups ◽  
Almost Simple Groups ◽  
Prime Graphs ◽  
A Finite Group ◽  
Nonabelian Composition Factor

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, p′ are joined by an edge if there is an element in G of order pp′. In [G. Y. Chen et al., Recognition of the finite almost simple groups PGL2(q) by their spectrum, Journal of Group Theory, 10 (2007) 71–85], it is proved that PGL(2, pk), where p is an odd prime and k > 1 is an integer, is recognizable by its spectrum. It is proved that if p > 19 is a prime number which is not a Mersenne or Fermat prime and Γ(G) = Γ(PGL(2, p)), then G has a unique nonabelian composition factor which is isomorphic to PSL(2, p). In this paper as the main result, we show that if p is an odd prime and k > 1 is an odd integer, then PGL(2, pk) is uniquely determined by its prime graph and so these groups are characterizable by their prime graphs.

Representations of Lie Groups by Contact Transformations, II: Non-Compact Simple Groups

1993 ◽  
Vol 45 (4) ◽  
pp. 778-802
Author(s):  
Carl Herz
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Compact Manifold ◽  
Transitive Group ◽  
Simple Groups ◽  
Representations Of Lie Groups ◽  
Contact Transformations

AbstractIf a Lie group acts faithfully as a transitive group of contact transformations of a compact manifold it is either compact with centre of dimension at most 1 or non-compact simple. The latter case is described

scholarly journals Number of Sylow subgroups in finite groups

2018 ◽  
Vol 21 (4) ◽  
pp. 695-712 ◽  
Author(s):  
Wenbin Guo ◽  
Evgeny P. Vdovin
Keyword(s):  
Finite Groups ◽  
Simple Groups ◽  
Alternative Proof ◽  
Sylow Subgroups ◽  
Almost Simple Groups

AbstractDenote by {\nu_{p}(G)} the number of Sylow p-subgroups of G. It is not difficult to see that {\nu_{p}(H)\leqslant\nu_{p}(G)} for {H\leqslant G}, however {\nu_{p}(H)} does not divide {\nu_{p}(G)} in general. In this paper we reduce the question whether {\nu_{p}(H)} divides {\nu_{p}(G)} for every {H\leqslant G} to almost simple groups. This result substantially generalizes the previous result by G. Navarro and also provides an alternative proof of Navarro’s 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).

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