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On groups of finite Morley rank with a split BN-pair of rank 1

Josh Wiscons
2011 Journal of Algebra  
We study groups of finite Morley rank with a split BN-pair of Tits rank 1 in the case where the normal complement to B ∩ N in B is infinite and abelian.  ...  In particular, we show that SL 2 (F ) and PSL 2 (F ) are the only infinite quasisimple L * -groups of finite Morley rank possessing a split BN-pair of Tits rank 1 where the normal complement to B ∩ N in  ...  The author would also like to acknowledge Jeffrey Burdges and Alexandre Borovik for their assistance with a couple of points as well as the referee for many helpful suggestions.  ... 
doi:10.1016/j.jalgebra.2010.12.020 fatcat:glxlvvnd3jdotdmlvjc6o4kcn4

Page 8654 of Mathematical Reviews Vol. , Issue 2004k [page]

2004 Mathematical Reviews  
Van Maldeghem on classifying groups of finite Morley rank having some BN-pair.  ...  Math. 21 (1973), 1-57; ibid. 24 (1974), 191-239; MR 50 #7335] for finite groups with split BN-pair of rank two to arbitrary groups without quoting this for the finite groups.  ... 

Page 1420 of Mathematical Reviews Vol. , Issue 96c [page]

1996 Mathematical Reviews  
definable BN-pair of Tits rank at least three.  ...  Fields of finite Morley rank are handled in Chapter 8.  ... 

Simple groups of finite morley rank and Tits buildings

Linus Kramer, Katrin Tent, Hendrik Van Maldeghem
1999 Israel Journal of Mathematics  
It follows that any infinite simple group of finite Morley rank with a spherical Moufang BN-pair of Tits rank 2 is either PSL3(K), PSp4(K ) or G2(K) for some algebraically closed field K.  ...  Using Theorem A we show THEOREM B: If G is an infinite simple group of finite Morley rank with a spherical Moufang BN-pair of Tits rank ~ 2, then G is (interpretably) isomorphic to a simple algebraic group  ...  The paper was written while the first two authors were guests at the Institute of Mathematics of The Hebrew University of Jerusalem.  ... 
doi:10.1007/bf02775036 fatcat:sum5rojxm5hrjdyrvmfm5uzzeu

Special abelian Moufang sets of finite Morley rank

Tom De Medts, Katrin Tent
2008 Journal of group theroy  
Moufang sets are split doubly transitive permutation groups, or equivalently, groups with a split BN-pair of rank one.  ...  In this paper, we study so-called special Moufang sets with abelian root groups, under the model-theoretic restriction that the groups have finite Morley rank.  ...  Acknowledgment This paper was written during a longer visit of the first author at the University of Bielefeld, supported by a travel grant from the Research Foundation in Flanders (Belgium) (F.W.O.  ... 
doi:10.1515/jgt.2008.040 fatcat:qfgyyy44nfcihaucn7zf34opha

Page 4190 of Mathematical Reviews Vol. , Issue 2000f [page]

2000 Mathematical Reviews  
In the paper under review the authors give a proof for the case of a simple group with a BN-pair of Tits rank at least 2 which is associated to a Moufang building.  ...  finite Morley rank and Tits buildings.  ... 

Classification of Simple K*-Groups of Finite Morley Rank and Even Type: Geometric Aspects [chapter]

Tuna Altinel, Alexandre V. Borovik, G. Cherlin
2003 Groups, Combinatorics and Geometry  
Generic groups can be handled either by constructing a BN -pair in G of (Tits) rank at least 3 [11] and the subsequent application of the classification of BN -pairs of finite Morley rank (Kramer, van  ...  Identification We apply a very general result from [18, Theorem 3.14] : Fact 4 Let G * be an infinite simple group of finite Morley rank with a spherical Moufang BN-pair of Tits rank 2.  ... 
doi:10.1142/9789812564481_0001 fatcat:7yxre46d75euzbxlvaiue2d3se

Moufang sets of finite Morley rank of odd type

Joshua Wiscons
2014 Journal of Algebra  
We show that for a wide class of groups of finite Morley rank the presence of a split BN-pair of Tits rank 1 forces the group to be of the form PSL_2 and the BN-pair to be standard.  ...  Specifically, we investigate infinite and so-called hereditarily proper Moufang sets of finite Morley rank in the case where the little projective group has no infinite elementary abelian 2-subgroups and  ...  Acknowledgments The author would like to thank Katrin Tent and Tuna Altınel for several valuable suggestions and comments that lead to a refinement of both the content and the presentation of the paper  ... 
doi:10.1016/j.jalgebra.2014.01.001 fatcat:rsidk6hdzjcw5iqy66ep22zf5q

Parabolic 2-local subgroups in groups of finite Morley rank of even type

Tuna Altınel, Alexandre Borovik, Gregory Cherlin, Luis-Jaime Corredor
2003 Journal of Algebra  
Thus A is finite and central in P . In particular [A, S] = 1. We claim that C(G, S) C(A). Then by the C(G, T )-theorem (Fact 16) and inspection of SL 2 (F ), we have A = 1, as claimed.  ...  We have P = N • (Q) with Q a nontrivial definable 2-subgroup of G. We suppose that P contains a Borel subgroup B of G and we set S = O 2 (B), which is connected by Fact 2. Let A = O 2 (P ).  ...  The point is that such extensions cannot have finite Morley rank. Fact 12 [BN, Theorem 8.4, p. 124] .  ... 
doi:10.1016/s0021-8693(03)00393-4 fatcat:dtjznxjdi5fappq3jggeoab4my

Page 1885 of Mathematical Reviews Vol. , Issue 2004c [page]

2004 Mathematical Reviews  
Tits [Buildings of spherical type and finite BN-pairs, Lecture Notes in Math., 386, Springer, Berlin, 1974; MR 57 #9866].  ...  This proves that there is no simple group of finite Morley rank of mixed type, provided the main conjecture is true for even type groups.  ... 

Page 4557 of Mathematical Reviews Vol. , Issue 95h [page]

1995 Mathematical Reviews  
The article is a first step towards a classification of BN-pairs of rank | and finite Morley rank. BN-pairs of rank | are doubly transitive permutation groups.  ...  We say that a finite group is a rational group if its irreducible complex characters are all rational-valued. Suppose now that G is 4557 a finit Witt | of G. which subgr Amo!  ... 

Special abelian Moufang sets of finite Morley rank in characteristic 2

Josh Wiscons
2010 Journal of group theroy  
In this paper we study special Moufang sets MðU; tÞ, with U abelian, under the additional restriction that they have finite Morley rank.  ...  endomorphisms of U centralize the Hua subgroup.  ...  It is shown in [10] that infinite simple groups of finite Morley rank with such a BN-pair of Tits rank at least 3 are indeed isomorphic to algebraic groups over algebraically closed fields.  ... 
doi:10.1515/jgt.2009.031 fatcat:xqzimkniozdvhdsavw7drgwivi

On small groups of finite Morley rank with a supertight automorphism [article]

Ulla Karhumäki, Pınar Uğurlu
2022 arXiv   pre-print
We consider an infinite simple group of finite Morley rank G of Prüfer 2-rank 1 which admits a supertight automorphism α such that the fixed-point subgroup C_G(α^n) is pseudofinite for all n∈ℕ∖{0}.  ...  In this set-up, we generalise the main theorem by dropping the assumption of the characteristic of the pseudofinite field F_1. We also correct a mistake from the earlier version.  ...  References 39 Acknowledgements We wish to thank Alexandre Borovik for bringing us together to work on this programme, for his encouragement and for his many helpful observations, especially related to finite  ... 
arXiv:2012.09582v3 fatcat:jlknslybmjaslbpts5satkfzmy

Page 6653 of Mathematical Reviews Vol. , Issue 2003i [page]

2003 Mathematical Reviews  
Morley rank.  ...  Pedersen (Binghamton, NY) See also 20093, 55005, 57024, 81192. 20 GROUP THEORY AND GENERALIZATIONS 20A Foundations 2003i:20001 20A15 03C60 20E42 Tent, Katrin (D-WRZB-IM; Wiirzburg) Split BN -pairs of finite  ... 

Book Review: The classification of quasithin groups I, II

Ronald Solomon
2005 Bulletin of the American Mathematical Society  
These volumes, classifying the quasithin finite simple groups of even characteristic, are a major milestone in the history of finite group theory.  ...  The classification of finite simple groups of odd type occupied scores of group theorists during the 1960's and 1970's.  ...  By Bender's Strongly Embedded Theorem, if M contains N G (X) for every non-identity subgroup X of T , then G is a split BN -pair of rank 1, in particular, a known simple group.  ... 
doi:10.1090/s0273-0979-05-01071-2 fatcat:3rawrfnhszgnzguzvojkoub3xq
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