scholarly journals Construction of Some Irreducible Subgroups of E8 and E6

2007 ◽  
Vol 10 ◽  
pp. 329-340 ◽  
Author(s):  
A.J.E Ryba
Keyword(s):  
Finite Groups ◽  
Interesting Property ◽  
Finite Subgroup ◽  
Group Of Lie Type ◽  
Minimal Module ◽  
Groups Of Lie Type ◽  
General Method

We construct two embeddings of finite groups into groups of Lie type. These embeddings have the interesting property that the finite subgroup acts irreducibly on a minimal module for the group of Lie type. We present our constructions as examples of a general method that obtains embeddings into groups of Lie type.

UNIPOTENT ELEMENTS IN REPRESENTATIONS OF FINITE GROUPS OF LIE TYPE

2012 ◽  
Vol 11 (02) ◽  
pp. 1250038 ◽  
Author(s):  
L. DI MARTINO ◽  
A. E. ZALESSKI
Keyword(s):  
Normal Subgroup ◽  
Irreducible Representation ◽  
Simple Group ◽  
Finite Groups ◽  
Closed Field ◽  
Group Of Lie Type ◽  
Algebraically Closed Field ◽  
Central Quotient ◽  
Representations Of Finite Groups ◽  
Groups Of Lie Type

Let G be a finite quasi-simple group of Lie type of defining characteristic r > 2. Let H = 〈h, G〉 be a group with normal subgroup G, where h is a non-central r-element of H. Let ϕ be an irreducible representation of H non-trivial on G over an algebraically closed field of characteristic ℓ ≠ r. We show that ϕ(h) has at least two distinct eigenvalues of multiplicity greater than 1, unless G is a central quotient of one of the following groups: SL(2, r), SL(2, 9) or Sp(4, 3), and H = G⋅Z(H).

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.

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