Some Ideas in the Classification of the Finite Simple Groups

2004 ◽  
pp. 725-731
Author(s):  
Gernot Stroth
Keyword(s):  
Simple Groups ◽  
Finite Simple Groups

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.

Variations on results on orders of products in finite groups

2020 ◽  
pp. 1-7
Author(s):  
Juan Martínez ◽  
Alexander Moretó
Keyword(s):  
Finite Groups ◽  
Prime Power ◽  
Simple Groups ◽  
Prime Power Order ◽  
Finite Simple Groups ◽  
Prime Divisors ◽  
Element Orders ◽  
Hall Subgroups ◽  
A Finite Group

In 2014, Baumslag and Wiegold proved that a finite group G is nilpotent if and only if o(xy) = o(x)o(y) for every x, y ∈ G with (o(x), o(y)) = 1. This has led to a number of results that characterize the nilpotence of a group (or the existence of nilpotent Hall subgroups, or the existence of normal Hall subgroups) in terms of prime divisors of element orders. Here, we look at these results with a new twist. The first of our main results asserts that G is nilpotent if and only if o(xy) ⩽ o(x)o(y) for every x, y ∈ G of prime power order with (o(x), o(y)) = 1. As an immediate consequence, we recover the Baumslag–Wiegold theorem. The proof of this result is elementary. We prove some variations of this result that depend on the classification of finite simple groups.

scholarly journals On a Theorem of P. Fong

2003 ◽  
Vol 171 ◽  
pp. 197-206
Author(s):  
Inna Korchagina
Keyword(s):  
Simple Groups ◽  
Finite Simple Groups

AbstractThis paper is a contribution to the “revision” project of Gorenstein, Lyons and Solomon, whose goal is to produce a unified proof of the Classification of Finite Simple Groups.

The Classification of the Finite Simple Groups, Number 6

2004 ◽  
Author(s):  
Daniel Gorenstein ◽  
Richard Lyons ◽  
Ronald Solomon
Keyword(s):  
Simple Groups ◽  
Finite Simple Groups

The Classification of Finite Simple Groups

2011 ◽  
Author(s):  
Michael Aschbacher ◽  
Richard Lyons ◽  
Stephen Smith ◽  
Ronald Solomon
Keyword(s):  
Simple Groups ◽  
Finite Simple Groups

The Classification of the Finite Simple Groups, Number 7

2018 ◽  
Author(s):  
Daniel Gorenstein ◽  
Richard Lyons ◽  
Ronald Solomon
Keyword(s):  
Simple Groups ◽  
Finite Simple Groups

scholarly journals Isospectral drums and simple groups

2018 ◽  
Vol 15 (04) ◽  
pp. 1850060
Author(s):  
Koen Thas
Keyword(s):  
Physical Properties ◽  
The Other ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Planar Domains ◽  
Other Hand ◽  
Isospectral Manifolds ◽  
Transplantation Technique ◽  
Rule Out

Nearly every known pair of isospectral but nonisometric manifolds — with as most famous members isospectral bounded [Formula: see text]-planar domains which makes one “not hear the shape of a drum” [M. Kac, Can one hear the shape of a drum? Amer. Math. Monthly 73(4 part 2) (1966) 1–23] — arise from the (group theoretical) Gassmann–Sunada method. Moreover, all the known [Formula: see text]-planar examples (so counter examples to Kac’s question) are constructed through a famous specialization of this method, called transplantation. We first describe a number of very general classes of length equivalent manifolds, with as particular cases isospectral manifolds, in each of the constructions starting from a given example that arises itself from the Gassmann–Sunada method. The constructions include the examples arising from the transplantation technique (and thus in particular the known planar examples). To that end, we introduce four properties — called FF, MAX, PAIR and INV — inspired by natural physical properties (which rule out trivial constructions), that are satisfied for each of the known planar examples. Vice versa, we show that length equivalent manifolds with FF, MAX, PAIR and INV which arise from the Gassmann–Sunada method, must fall under one of our prior constructions, thus describing a precise classification of these objects. Due to the nature of our constructions and properties, a deep connection with finite simple groups occurs which seems, perhaps, rather surprising in the context of this paper. On the other hand, our properties define in some sense physically irreducible pairs of length equivalent manifolds — “atoms” of general pairs of length equivalent manifolds, in that such a general pair of manifolds is patched up out of irreducible pairs — and that is precisely what simple groups are for general groups.

scholarly journals On the Modular Version of Ito’s Theorem on Character Degrees for Groups of Odd Order

1987 ◽  
Vol 105 ◽  
pp. 109-119 ◽  
Author(s):  
Olaf Manz
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Prime Number ◽  
Character Degrees ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Classical Representation ◽  
Odd Order ◽  
Complex Characters

One of the most useful theorems in classical representation theory is a result due to N. Ito, which can be stated using the classification of the finite simple groups in the following way.THEOREM (N. Ito, G. Michler). Let Irr (G) be the set of all irreducible complex characters of the finite group G and q be a prime number. Then if and only if G has a normal, abelian Sylow-q-subgroup.

scholarly journals THE -GENERATION OF THE SPECIAL LINEAR GROUPS OVER FINITE FIELDS

2016 ◽  
Vol 95 (1) ◽  
pp. 48-53 ◽  
Author(s):  
MARCO ANTONIO PELLEGRINI
Keyword(s):  
Finite Fields ◽  
Simple Groups ◽  
Linear Groups ◽  
Finite Simple Groups ◽  
Special Linear ◽  
Special Linear Groups

We complete the classification of the finite special linear groups $\text{SL}_{n}(q)$ which are $(2,3)$-generated, that is, which are generated by an involution and an element of order $3$. This also gives the classification of the finite simple groups $\text{PSL}_{n}(q)$ which are $(2,3)$-generated.

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