On 2-Groups With No Normal Abelian Subgroups of Rank 3, and Their Occurrence as Sylow 2-Subgroups of Finite Simple Groups

1970 ◽  
Vol 150 (2) ◽  
pp. 345 ◽  
Author(s):  
Anne R. MacWilliams
Keyword(s):  
Simple Groups ◽  
Finite Simple Groups ◽  
Abelian Subgroups

scholarly journals Centralizers of abelian subgroups in locally finite simple groups

1997 ◽  
Vol 40 (2) ◽  
pp. 217-225
Author(s):  
M. Kuzucuoǧlu
Keyword(s):  
Simple Group ◽  
Finite Length ◽  
Finite Simple Group ◽  
Abelian Subgroup ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Locally Finite ◽  
Odd Order ◽  
Non Linear ◽  
Abelian Subgroups

It is shown that, if a non-linear locally finite simple group is a union of finite simple groups, then the centralizer of every element of odd order has a series of finite length with factors which are either locally solvable or non-abelian simple. Moreover, at least one of the factors is non-linear simple. This is also extended to abelian subgroup of odd orders.

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) .

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
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