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 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 Small Degree Representations of Finite Chevalley Groups in Defining Characteristic

2001 ◽  
Vol 4 ◽  
pp. 135-169 ◽  
Author(s):  
Frank Lübeck
Keyword(s):  
Algebraic Groups ◽  
Small Degree ◽  
Irreducible Representations ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Linear Algebraic Groups ◽  
Large Rank ◽  
Projective Representations ◽  
Computer Calculations ◽  
Simply Connected

AbstractThe author has determined, for all simple simply connected reductive linear algebraic groups defined over a finite field, all the irreducible representations in their defining characteristic of degree below some bound. These also give the small degree projective representations in defining characteristic for the corresponding finite simple groups. For large rank l, this bound is proportional to l3, and for rank less than or equal to 11 much higher. The small rank cases are based on extensive computer calculations.

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.

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.

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