The P-radical classes in simple algebraic groups and finite groups of Lie type

2012 ◽  
Vol 15 (5) ◽  
Author(s):  
R. Lawther
Keyword(s):  
Finite Group ◽  
Algebraic Group ◽  
Parabolic Subgroup ◽  
Finite Groups ◽  
Algebraic Groups ◽  
Unipotent Radical ◽  
Maximal Parabolic Subgroup ◽  
Simple Algebraic Group ◽  
A Finite Group ◽  
Groups Of Lie Type

Abstract.Given either a simple algebraic group or a finite group of Lie type, of rank at least 2, and a maximal parabolic subgroup, we determine which non-trivial unipotent classes have the property that their intersection with the parabolic subgroup is contained within its unipotent radical. Such classes are rare; listing them provides a basis for inductive arguments.

scholarly journals Generalized q-Schur algebras and modular representation theory of finite groups with split (BN)-pairs

1999 ◽  
Vol 1999 (511) ◽  
pp. 145-191 ◽  
Author(s):  
Richard Dipper ◽  
Jochen Gruber
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Parabolic Type ◽  
Modular Representation ◽  
Weyl Groups ◽  
Modular Representation Theory ◽  
Schur Algebras ◽  
Schur Algebra ◽  
A Finite Group ◽  
Groups Of Lie Type

Abstract We introduce a generalized version of a q-Schur algebra (of parabolic type) for arbitrary Hecke algebras over extended Weyl groups. We describe how the decomposition matrix of a finite group with split BN-pair, with respect to a non-describing prime, can be partially described by the decomposition matrices of suitably chosen q-Schur algebras. We show that the investigated structures occur naturally in finite groups of Lie type.

scholarly journals Permutation characters of finite groups of Lie type

1979 ◽  
Vol 27 (3) ◽  
pp. 378-384 ◽  
Author(s):  
David B. Surowski
Keyword(s):  
Fixed Points ◽  
Algebraic Group ◽  
Weyl Group ◽  
Parabolic Subgroup ◽  
Finite Groups ◽  
Linear Algebraic Group ◽  
Finite Subgroup ◽  
Semisimple Element ◽  
Frobenius Endomorphism ◽  
Groups Of Lie Type

AbstractLet g be a connected reductive linear algebraic group, and let G = gσ be the finite subgroup of fixed points, where σ is the generalized Frobenius endomorphism of g. Let x be a regular semisimple element of G and let w be a corresponding element of the Weyl group W. In this paper we give a formula for the number of right cosets of a parabolic subgroup of G left fixed by x, in terms of the corresponding action of w in W. In case G is untwisted, it turns out thta x fixes exactly as many cosets as does W in the corresponding permutation representation.

VOLUME-PRESERVING ACTIONS OF SIMPLE ALGEBRAIC ℚ-GROUPS ON LOW-DIMENSIONAL MANIFOLDS

2012 ◽  
Vol 04 (02) ◽  
pp. 115-120
Author(s):  
DAVE WITTE MORRIS ◽  
ROBERT J. ZIMMER
Keyword(s):  
Finite Group ◽  
Algebraic Group ◽  
Compact Manifold ◽  
Dimensional Manifold ◽  
Simple Algebraic Group ◽  
Rank 2 ◽  
Low Dimensional ◽  
A Finite Group ◽  
Volume Preserving

We prove that SL (n, ℚ) has no nontrivial, C∞, volume-preserving action on any compact manifold of dimension strictly less than n. More generally, suppose G is a connected, isotropic, almost-simple algebraic group over ℚ, such that the simple factors of every localization of G have rank ≥ 2. If there does not exist a nontrivial homomorphism from G(ℝ)° to GL (d, ℂ), then every C∞, volume-preserving action of G(ℚ) on any compact d-dimensional manifold must factor through a finite group.

scholarly journals Unipotent elements forcing irreducibility in linear algebraic groups

2018 ◽  
Vol 21 (3) ◽  
pp. 365-396 ◽  
Author(s):  
Mikko Korhonen
Keyword(s):  
Algebraic Group ◽  
Parabolic Subgroup ◽  
Algebraic Groups ◽  
Closed Field ◽  
Unipotent Element ◽  
Tilting Modules ◽  
Simple Algebraic Group ◽  
Algebraically Closed Field ◽  
Linear Algebraic Groups ◽  
Highest Weight

Abstract Let G be a simple algebraic group over an algebraically closed field K of characteristic {p>0} . We consider connected reductive subgroups X of G that contain a given distinguished unipotent element u of G. A result of Testerman and Zalesski [D. Testerman and A. Zalesski, Irreducibility in algebraic groups and regular unipotent elements, Proc. Amer. Math. Soc. 141 2013, 1, 13–28] shows that if u is a regular unipotent element, then X cannot be contained in a proper parabolic subgroup of G. We generalize their result and show that if u has order p, then except for two known examples which occur in the case {(G,p)=(C_{2},2)} , the subgroup X cannot be contained in a proper parabolic subgroup of G. In the case where u has order {>p} , we also present further examples arising from indecomposable tilting modules with quasi-minuscule highest weight.

scholarly journals A Coincidence Theorem for Holomorphic Maps toG/P

2003 ◽  
Vol 46 (2) ◽  
pp. 291-298 ◽  
Author(s):  
Parameswaran Sankaran
Keyword(s):  
Algebraic Group ◽  
Parabolic Subgroup ◽  
Holomorphic Maps ◽  
Projective Varieties ◽  
Maximal Parabolic Subgroup ◽  
Simple Algebraic Group ◽  
Coincidence Theorem ◽  
Complex Projective

AbstractThe purpose of this note is to extend to an arbitrary generalized Hopf and Calabi-Eckmann manifold the following result of Kalyan Mukherjea: Letdenote a Calabi-Eckmann manifold. If f, g : Vn→are any two holomorphic maps, at least one of them being non-constant, then there exists a coincidence: f(x) = g(x) for some x ∈ Vn. Our proof involves a coincidence theorem for holomorphic maps to complex projective varieties of the formG/PwhereGis complex simple algebraic group andP⊂Gis a maximal parabolic subgroup, where one of the maps is dominant.

scholarly journals Elements with Square Roots in Finite Groups

2005 ◽  
Vol 12 (04) ◽  
pp. 677-690 ◽  
Author(s):  
M. S. Lucido ◽  
M. R. Pournaki
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Square Root ◽  
Alternating Groups ◽  
Square Roots ◽  
A Finite Group ◽  
Groups Of Lie Type

In this paper, we study the probability that a randomly chosen element in a finite group has a square root, in particular the simple groups of Lie type of rank 1, the sporadic finite simple groups and the alternating groups.

Recognizing Finite Groups Through Order and Degree Pattern

2008 ◽  
Vol 15 (03) ◽  
pp. 449-456 ◽  
Author(s):  
A. R. Moghaddamfar ◽  
A. R. Zokayi
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Symmetric Groups ◽  
Simple Groups ◽  
Alternating Groups ◽  
Degree Pattern ◽  
Sporadic Simple Groups ◽  
A Finite Group ◽  
Groups Of Lie Type

The degree pattern of a finite group G is introduced in [10] and it is proved that the following simple groups are uniquely determined by their degree patterns and orders: all sporadic simple groups, alternating groups Ap (p ≥ 5 is a twin prime) and some simple groups of Lie type. In this paper, we continue this investigation. In particular, we show that the automorphism groups of sporadic simple groups (except Aut (J2) and Aut (McL)), all simple C22-groups, the alternating groups Ap, Ap+1, Ap+2 and the symmetric groups Sp, Sp+1, where p is a prime, are also uniquely determined by their degree patterns and orders.

scholarly journals Finite groups with some weakly pronormal subgroups

2020 ◽  
Vol 18 (1) ◽  
pp. 1742-1747
Author(s):  
Jianjun Liu ◽  
Mengling Jiang ◽  
Guiyun Chen
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
A Finite Group

Abstract A subgroup H of a finite group G is called weakly pronormal in G if there exists a subgroup K of G such that G = H K G=HK and H ∩ K H\cap K is pronormal in G. In this paper, we investigate the structure of the finite groups in which some subgroups are weakly pronormal. Our results improve and generalize many known results.

scholarly journals A on simplicity criterion for finite groups

1969 ◽  
Vol 10 (3-4) ◽  
pp. 359-362
Author(s):  
Nita Bryce
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Odd Order ◽  
A Finite Group

M. Suzuki [3] has proved the following theorem. Let G be a finite group which has an involution t such that C = CG(t) ≅ SL(2, q) and q odd. Then G has an abelian odd order normal subgroup A such that G = CA and C ∩ A = 〈1〉.

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