scholarly journals Almost cyclic elements in cross-characteristic representations of finite groups of Lie type

2020 ◽  
Vol 23 (2) ◽  
pp. 235-285
Author(s):  
Lino Di Martino ◽  
Marco A. Pellegrini ◽  
Alexandre E. Zalesski
Keyword(s):  
Significant Contribution ◽  
Automorphism Groups ◽  
Irreducible Representations ◽  
Simple Groups ◽  
Closed Field ◽  
General Picture ◽  
Weil Representations ◽  
Representations Of Finite Groups ◽  
Finite Classical Groups ◽  
Groups Of Lie Type

AbstractThis paper is a significant contribution to a general programme aimed to classify all projective irreducible representations of finite simple groups over an algebraically closed field, in which the image of at least one element is represented by an almost cyclic matrix (that is, a square matrix M of size n over a field {\mathbb{F}} with the property that there exists {\alpha\in\mathbb{F}} such that M is similar to {\operatorname{diag}(\alpha\cdot\mathrm{Id}_{k},M_{1})}, where {M_{1}} is cyclic and {0\leq k\leq n}). While a previous paper dealt with the Weil representations of finite classical groups, which play a key role in the general picture, the present paper provides a conclusive answer for all cross-characteristic projective irreducible representations of the finite quasi-simple groups of Lie type and their automorphism groups.

scholarly journals On the Steinberg Character of a Finite Simple Group of Lie Type

1971 ◽  
Vol 12 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Bhama Srinivasan
Keyword(s):  
Algebraic Group ◽  
Linear Algebraic Group ◽  
Simple Groups ◽  
Closed Field ◽  
Finite Simple Groups ◽  
Reductive Algebraic Group ◽  
Simple Algebraic Group ◽  
Simply Connected ◽  
Groups Of Lie Type ◽  
Steinberg Character

Let K be an algebraically closed field of characteristic ρ >0. If G is a connected, simple connected, semisimple linear algebraic group defined over K and σ an endomorphism of G onto G such that the subgroup Gσ of fixed points of σ is finite, Steinberg ([6] [7]) has shown that there is a complex irreducible character χ of Gσ with the following properties. χ vanishes at all elements of Gσ which are not semi- simple, and, if x ∈ G is semisimple, χ(x) = ±n(x) where n(x)is the order of a Sylow p-subgroup of (ZG(x))σ (ZG(x) is the centraliser of x in G). If G is simple he has, in [6], identified the possible groups Gσ they are the Chevalley groups and their twisted analogues over finite fields, that is, the ‘simply connected’ versions of finite simple groups of Lie type. In this paper we show, under certain restrictions on the type of the simple algebraic group G an on the characteristic of K, that χ can be expressed as a linear combination with integral coefficients of characters induced from linear characters of certain naturally defined subgroups of Gσ. This expression for χ gives an explanation for the occurence of n(x) in the formula for χ (x), and also gives an interpretation for the ± 1 occuring in the formula in terms of invariants of the reductive algebraic group ZG(x).

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).

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 Almost Simple Groups of Lie Type and Symmetric Designs with $\lambda$ Prime

10.37236/9366 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Seyed Hassan Alavi ◽  
Mohsen Bayat ◽  
Ashraf Daneshkhah
Keyword(s):  
Projective Space ◽  
Simple Group ◽  
Finite Simple Group ◽  
Automorphism Groups ◽  
Simple Groups ◽  
Almost Simple Group ◽  
Group Of Lie Type ◽  
Symmetric Designs ◽  
Almost Simple Groups ◽  
Groups Of Lie Type

In this article, we investigate symmetric $(v,k,\lambda)$ designs $\mathcal{D}$ with $\lambda$ prime admitting flag-transitive and point-primitive automorphism groups $G$. We prove that if $G$ is an almost simple group with socle a finite simple group of Lie type, then $\mathcal{D}$ is either the point-hyperplane design of a projective space $\mathrm{PG}_{n-1}(q)$, or it is of parameters  $(7,4,2)$, $(11,5,2)$, $(11,6,3)$ or $(45,12,3)$.

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.

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