Applications of character theory of finite simple groups

2018 ◽  
pp. 323-352 ◽  
Author(s):  
Martin Liebeck
Keyword(s):  
Simple Groups ◽  
Character Theory ◽  
Finite Simple Groups

scholarly journals Conjugacy in groups with dihedral 3-normalisers

1977 ◽  
Vol 18 (2) ◽  
pp. 167-173 ◽  
Author(s):  
N. K. Dickson
Keyword(s):  
Conjugacy Class ◽  
The Other ◽  
Simple Groups ◽  
Character Theory ◽  
Finite Simple Groups ◽  
Strong Connection

Much work has been carried out on the classification of finite simple groups in terms of the structures of centralisers of involutions. However, it is sometimes the case that these classification results cannot be applied to particular problems even although information is available about one conjugacy class of involutions. The trouble is that information about the other classes can be almost non-existent. In this paper we deal with a situation where character theory can be employed to give a strong connection between the orders of centralisers of different classes of involutions, enabling information about one class to be used to give information about other classes. We prove the following result.

scholarly journals Finite groups with an automorphism inverting, squaring or cubing a non-negligible fraction of elements

2019 ◽  
Vol 18 (03) ◽  
pp. 1950055 ◽  
Author(s):  
Alexander Bors
Keyword(s):  
Finite Groups ◽  
Classical Result ◽  
Simple Groups ◽  
Character Theory ◽  
Finite Simple Groups ◽  
Long Time ◽  
Commuting Probability

Finite groups with an automorphism mapping a sufficiently large proportion of elements to their inverses, squares and cubes have been studied for a long time, and the gist of the results on them is that they are “close to being abelian”. In this paper, we consider finite groups [Formula: see text] such that, for a fixed but arbitrary [Formula: see text], some automorphism of [Formula: see text] maps at least [Formula: see text] many elements of [Formula: see text] to their inverses, squares and cubes. We will relate these conditions to some parameters that measure, intuitively speaking, how far the group [Formula: see text] is from being solvable, nilpotent or abelian; most prominently the commuting probability of [Formula: see text], i.e. the probability that two independently uniformly randomly chosen elements of [Formula: see text] commute. To this end, we will use various counting arguments, the classification of the finite simple groups and some of its consequences, as well as a classical result from character theory.

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
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