Note on a consequence for affine groups of the classification theorem for finite simple groups

1983 ◽  
Vol 14 (1) ◽  
Author(s):  
JenniferD. Key
Keyword(s):  
Classification Theorem ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Affine Groups

The uniqueness of envelopes in ℵ0-categorical, ℵ0-stable structures

1984 ◽  
Vol 49 (4) ◽  
pp. 1171-1184 ◽  
Author(s):  
James Loveys
Keyword(s):  
Group Theory ◽  
Finite Subset ◽  
Classification Theorem ◽  
Simple Groups ◽  
Finite Model ◽  
Finite Simple Groups ◽  
Morley Rank ◽  
Categorical Theory ◽  
Rank 2 ◽  
Stable Structures

The Classification Theorem for ℵ0-categorical strictly minimal sets says that if H is strictly minimal and ℵ0-categorical, either H has in effect no structure at all or is essentially an affine or projective space over a finite field. Zil′ber, in [Z2], showed that if H were a counterexample to this Classification Theorem it would interpret a rank 2, degree 1 pseudoplane. Cherlin later noticed (see [CHL, Appendices 2 and 3], for the proof) that the Classification Theorem is a consequence of the Classification Theorem for finite simple groups. In [Z4] and [Z5], Zil′ber found a quite different proof of the Classification Theorem using no deep group theory.Meanwhile in [Z3], Zil′ber introduced the notion of envelope in an attempt to prove that no complete totally categorical theory T can be finitely axiomatizable. The idea of the proof was to show that if M is a model of such a T and H ⊆ M is strongly minimal, then an envelope of any sufficiently large finite subset of H is a finite model of any fixed finite subset of T. [Z3] contains an error, which Zil′ber has since corrected (in a nontrivial way).In [CHL], Cherlin, Harrington and Lachlan used the Classification Theorem to expand and reorganize Zil′ber's work. In particular, they generalized most of his work to ℵ0-categorical, ℵ0-stable structures, proved the Morley rank is finite in these structures, and introduced the powerful Coordinatization Theorem (Theorem 3.1 of [CHL]; Proposition 1.14 of the present paper). They also showed that ℵ0-categorical, ℵ0-stable structures are not finitely axiomatizable using a notion of envelope that is the same as Zil′ber's except in one particularly perverse case; [CHL]'s notion of envelope is used throughout the current paper. Peretyat'kin [P] has found an example of an ℵ1-categorical finitely axiomatizable structure.

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

scholarly journals Periodic Groups Saturated with Finite Simple Groups of Lie Type of Rank 1

2018 ◽  
Vol 57 (1) ◽  
pp. 81-86
Author(s):  
A. A. Shlepkin
Keyword(s):  
Simple Groups ◽  
Finite Simple Groups ◽  
Periodic Groups ◽  
Groups Of Lie Type
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