Fusion of 2-elements in groups of finite Morley rank

2001 ◽  
Vol 66 (2) ◽  
pp. 722-730
Author(s):  
Luis-Jaime Corredor
Keyword(s):  
Algebraic Groups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Morley Rank ◽  
Finite Morley Rank ◽  
Abelian Subgroups ◽  
Characteristic 2

The Alperin-Goldschmidt Fusion Theorem [1, 5], when combined with pushing up [7], was a useful tool in the classification of the finite simple groups. Similar theorems are needed in the study of simple groups of finite Morley rank, in the even type case (that is, when the Sylow 2-subgroups are of bounded exponent, as in algebraic groups over fields of characteristic 2). In that context a body of results relating to fusion of 2-elements and the structure of 2-local subgroups is needed: pushing up, and the classification of groups with strongly or weakly embedded subgroups, or have strongly closed abelian subgroups (c.f, [2]). Two theorems of Alperin-Goldschmidt type are proved here. Both are needed in applications.The following is an exact analog of the Alperin-Goldschmidt Fusion Theorem for groups of finite Morley rank, in the case of 2-elements:Theorem 1.1. Let G be a group of finite Morley rank, and P a Sylow 2-subgroup of G. If A, B ⊆ P are conjugate in G, then there are subgroups Hi ≤ Pand elementsxi ∈ N(Hi) for 1 ≤ i ≤ n, and an elementy ∈ N(P), such that for all i:1. Hi is a tame intersection of two Sylow 2-subgroups;2. CP(Hi) ≤ Hi;3. N(Hi)/Hiis 2-isolatedand(a) (b) .

Groups of finite Morley rank with transitive group automorphisms

1989 ◽  
Vol 54 (3) ◽  
pp. 1080-1082
Author(s):  
Ali Nesin
Keyword(s):  
Finite Order ◽  
Short Note ◽  
Algebraic Groups ◽  
Transitive Group ◽  
Simple Groups ◽  
Morley Rank ◽  
Nontrivial Center ◽  
Finite Morley Rank ◽  
Group Automorphisms

The aim of this short note is to prove the following result:Theorem. Let G be a group of finite Morley rank with Aut G acting transitively on G/{1}. Then G is either abelian or a bad group.Bad groups were first defined by Cherlin [Ch]: these are groups of finite Morley rank without solvable and nonnilpotent connected subgroups. They have been investigated by the author [Ne 1], Borovik [Bo], Corredor [Co], and Poizat and Borovik [Bo-Po]. They are not supposed to exist, but we are far from proving their nonexistence. This is one of the major obstacles to proving Cherlin's conjecture: infinite simple groups of finite Morley rank are algebraic groups.If the group G of the theorem is finite, then it is well known that G ≈ ⊕Zp for some prime p: clearly all elements of G have the same order, say p, a prime. Thus G is a finite p-group, so has a nontrivial center. But Aut G acts transitively; thus G is abelian. Since it has exponent p, G ≈ ⊕Zp.The same proof for infinite G does not work even if it has finite Morley rank, for the following reasons:1) G may not contain an element of finite order.2) Even if G does contain an element of finite order, i.e. if G has exponent p, we do not know if G must have a nontrivial center.

scholarly journals CONJUGACY OF CARTER SUBGROUPS IN GROUPS OF FINITE MORLEY RANK

2008 ◽  
Vol 08 (01) ◽  
pp. 41-92 ◽  
Author(s):  
OLIVIER FRÉCON
Keyword(s):  
Conjugacy Class ◽  
Algebraic Groups ◽  
Simple Groups ◽  
Morley Rank ◽  
Minimal Counterexample ◽  
Finite Morley Rank ◽  
Cartan Subgroups

The Cherlin–Zil'ber Conjecture states that all simple groups of finite Morley rank are algebraic. We prove that any minimal counterexample to this conjecture has a unique conjugacy class of Carter subgroups, which are analogous to Cartan subgroups in algebraic groups.

Genericity, generosity, and tori

2008 ◽  
Vol 7 (4) ◽  
pp. 705-722 ◽  
Author(s):  
Gregory Cherlin
Keyword(s):  
Group Theory ◽  
Geometric Structure ◽  
Simple Groups ◽  
Morley Rank ◽  
Finite Morley Rank ◽  
Classification Project

AbstractWhile the classification project for the simple groups of finite Morley rank is unlikely to produce a classification of the simple groups of finite Morley rank, the enterprise has already arrived at a considerably closer approximation to that ideal goal than could have been realistically anticipated, with a mix of results of several flavors, some classificatory and others more structural, which can be combined when the stars are suitably aligned to produce results at a level of generality which, in parallel areas of group theory, would normally require either some additional geometric structure, or an explicit classification. And Bruno Poizat is generally awesome, though sometimes he goes too far.

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.

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.

scholarly journals Tame minimal simple groups of finite Morley rank

2004 ◽  
Vol 276 (1) ◽  
pp. 13-79 ◽  
Author(s):  
Gregory Cherlin ◽  
Eric Jaligot
Keyword(s):  
Simple Groups ◽  
Morley Rank ◽  
Finite Morley Rank

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.

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