Finite simple groups in which a Sylow 2-subgroup is an extension of an Abelian group by a group of rank 1

1975 ◽  
Vol 14 (3) ◽  
pp. 179-187 ◽  
Author(s):  
A. S. Kondrat'ev
Keyword(s):  
Abelian Group ◽  
Simple Groups ◽  
Finite Simple Groups

On the non-commuting graph in finite Moufang loops

2018 ◽  
Vol 17 (04) ◽  
pp. 1850070
Author(s):  
Karim Ahmadidelir
Keyword(s):  
Abelian Group ◽  
Nilpotent Groups ◽  
Moufang Loop ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Commutative Moufang Loop ◽  
Commuting Graph ◽  
Moufang Loops ◽  
Central Elements ◽  
Vertex Set

The non-commuting graph associated to a non-abelian group [Formula: see text], [Formula: see text], is a graph with vertex set [Formula: see text] where distinct non-central elements [Formula: see text] and [Formula: see text] of [Formula: see text] are joined by an edge if and only if [Formula: see text]. The non-commuting graph of a non-abelian finite group has received some attention in existing literature. Recently, many authors have studied the non-commuting graph associated to a non-abelian group. In particular, the authors put forward the following conjectures: Conjecture 1. Let [Formula: see text] and [Formula: see text] be two non-abelian finite groups such that [Formula: see text]. Then [Formula: see text]. Conjecture 2 (AAM’s Conjecture). Let [Formula: see text] be a finite non-abelian simple group and [Formula: see text] be a group such that [Formula: see text]. Then [Formula: see text]. Some authors have proved the first conjecture for some classes of groups (specially for all finite simple groups and non-abelian nilpotent groups with irregular isomorphic non-commuting graphs) but in [Moghaddamfar, About noncommuting graphs, Sib. Math. J. 47(5) (2006) 911–914], Moghaddamfar has shown that it is not true in general with some counterexamples to this conjecture. On the other hand, Solomon and Woldar proved the second conjecture, in [R. Solomon and A. Woldar, Simple groups are characterized by their non-commuting graph, J. Group Theory 16 (2013) 793–824]. In this paper, we will define the same concept for a finite non-commutative Moufang loop [Formula: see text] and try to characterize some finite non-commutative Moufang loops with their non-commuting graph. Particularly, we obtain examples of finite non-associative Moufang loops and finite associative Moufang loops (groups) of the same order which have isomorphic non-commuting graphs. Also, we will obtain some results related to the non-commuting graph of a finite non-commutative Moufang loop. Finally, we give a conjecture stating that the above result is true for all finite simple Moufang loops.

scholarly journals A diameter bound for finite simple groups of large rank

2017 ◽  
Vol 95 (2) ◽  
pp. 455-474 ◽  
Author(s):  
Arindam Biswas ◽  
Yilong Yang
Keyword(s):  
Simple Groups ◽  
Finite Simple Groups ◽  
Large Rank

scholarly journals New Beauville surfaces and finite simple groups

2013 ◽  
Vol 142 (3-4) ◽  
pp. 391-408 ◽  
Author(s):  
Shelly Garion ◽  
Matteo Penegini
Keyword(s):  
Simple Groups ◽  
Finite Simple Groups ◽  
Beauville Surfaces

scholarly journals Finite simple exceptional groups of Lie type in which all subgroups of odd index are pronormal

2020 ◽  
Vol 23 (6) ◽  
pp. 999-1016
Author(s):  
Anatoly S. Kondrat’ev ◽  
Natalia V. Maslova ◽  
Danila O. Revin
Keyword(s):  
Simple Groups ◽  
Finite Simple Groups ◽  
Exceptional Groups ◽  
Odd Index ◽  
Groups Of Lie Type

AbstractA subgroup H of a group G is said to be pronormal in G if H and {H^{g}} are conjugate in {\langle H,H^{g}\rangle} for every {g\in G}. In this paper, we determine the finite simple groups of type {E_{6}(q)} and {{}^{2}E_{6}(q)} in which all the subgroups of odd index are pronormal. Thus, we complete a classification of finite simple exceptional groups of Lie type in which all the subgroups of odd index are pronormal.

On Pronormal Subgroups in Finite Simple Groups

2018 ◽  
Vol 98 (2) ◽  
pp. 405-408 ◽  
Author(s):  
A. S. Kondrat’ev ◽  
N. V. Maslova ◽  
D. O. Revin
Keyword(s):  
Simple Groups ◽  
Finite Simple Groups

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.

A new characterization of some finite simple groups

2015 ◽  
Vol 56 (1) ◽  
pp. 78-82 ◽  
Author(s):  
M. F. Ghasemabadi ◽  
A. Iranmanesh ◽  
F. Mavadatpour
Keyword(s):  
Simple Groups ◽  
Finite Simple Groups

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].

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