Isomorphy classes of finite order automorphisms of SL(2,k)

In this paper, we consider the order m k-automorphisms of SL(2,k). We first characterize the forms that order m k-automorphisms of SL(2,k) take and then we find simple conditions on matrices A and B, involving eigenvalues and the field that the entries of A and B lie in, that are equivalent to isomorphy between the order m k-automorphisms Inn_A and Inn_B. We examine the number of isomorphy classes and conclude with examples for selected fields.     

On B(n,k) 2-groups

A finite group G is said to be a B(n, k) group if for any n-element subset {a ₁,…, a _n } of G, |{a _i a _j |1 ≤ i, j ≤ n}| ≤k. In this article, we give characterizations of the B(5, 19) 2-groups, and the B(6, k) 2-groups for 21 ≤ k ≤ 28.     

On the Classification of Orbits of Symmetric Subgroups Acting on Flag Varieties of SL(2, k)

Symmetric k-varieties are a generalization of symmetric spaces to general fields. Orbits of a minimal parabolic k-subgroup acting on a symmetric k-variety are essential in the study of symmetric k-varieties and their representations. In this article, we present the classification of these orbits for the group SL(2,k) for a number of base fields k, including finite fields and the 𝔭-adic numbers. We use the characterization in Helminck and Wang (1993 Helminck , A. G. , Wang , S. P. ( 1993 ). On rationality properties of involutions of reductive groups . Adv. Math. 99 ( 1 ): 26 – 96 .[Crossref], [Web of Science ®] , [Google Scholar]), which requires one to first classify the orbits of the θ-stable maximal k-split tori under the action of the k-points of the fixed point group.     

2-Cohomologies of the groups SL(n, q)

Let G = SL(n, q), where q is odd, V be a natural module over G, and L = S²(V) be its symmetric square. We construct a 2-cohomology group H²(G, L). The group is one-dimensional over F _q if n = 2 and q ≠ 3, and also if (n, q) = (4, 3). In all other cases H²(G, L) = 0. Previously, such groups H²(G, L) were known for the cases where n = 2 or q = p is prime. We state that H²(G, L) are trivial for n ⩾ 3 and q = p^m, m ⩾ 2. In proofs, use is made of rather elementary (noncohomological) methods. __________ Translated from Algebra i Logika, Vol. 47, No. 6, pp. 687–704, November–December, 2008.     

Cosine transforms over fields of characteristic 2

In this paper, we introduce cosine transforms over fields of characteristic 2. Our approach complements previous definitions of finite field trigonometric transforms, which only hold for fields whose characteristic is an odd prime. Besides introducing some new concepts related to trigonometry in finite fields, we discuss the eigenstructure and other important properties of the proposed transforms.     

Similarity over SL(n, F)

Let F be a finite field. It is shown that if AB axe n × n matrices with entries from F which are similar over GL(n, F), then AB are similar over SL(n, F), provided that some elementary divisor of xl- A is irreducible over F. The result remains true if F is any field such that any element of F may be represented as the norm of an element of any finite algebraic extension of F.     

The Triangulated Hopf Category n +SL(2)

We discuss an example of a triangulated Hopf category related to SL(2). It is an equivariant derived category equipped with multiplication and comultiplication functors and structure isomorphisms. We prove some coherence equations for structure isomorphisms. In particular, the Hopf category is monoidal.     

SL（2，q）的一个特征性质

本文证明了非单群系列ＳＬ（２，ｑ）（ｑ＝ｐ￣ｎ＞３）可以仅用其极大子群阶之集来刻划，从而得到了ＳＬ（２，ｑ）的一个特征性质．     

The symmetric (2k,k)‐graphs

A noncomplete graph G is called an (n, k)‐graph if it is n‐connected and G − X is not (n − |X| + 1)‐connected for any X ⊆ V(G) with |X| ≤ k. Mader conjectured that for k ≥ 3 the graph K_{2k + 2} − (1‐factor) is the unique (2k, k)‐graph. We settle this conjecture for strongly regular graphs, for edge transitive graphs, and for vertex transitive graphs. © 2000 John Wiley & Sons, Inc. J Graph Theory 36: 35–51, 2001     

-equivalence in adjoint classical groups over fields of virtual cohomological dimension

Let be a field of characteristic not whose virtual cohomological dimension is at most . Let be a semisimple group of adjoint type defined over . Let denote the normal subgroup of consisting of elements -equivalent to identity. We show that if is of classical type not containing a factor of type , . If is a simple classical adjoint group of type , we show that if and its multi-quadratic extensions satisfy strong approximation property, then . This leads to a new proof of the -triviality of -rational points of adjoint classical groups defined over number fields.

     

旗传递点本原且基柱为${\rm PSL}_n(q)$的$(v,k,4)$-对称设计

若$\cal D$为一个非平凡旗传递点本原对称$(v,k,4)$设计, 其基柱为${\rm PSL}_n(q)$且$G\leq {\rm Aut}(\cal D)$. 那么, $\cal D$ 必为$2$-$(15,8,4)$设计且${\rm Soc}(G)={\rm PSL}_2(9)$.     

On generalizations of Fermat curves over finite fields and their automorphisms

Let 𝒳 be an irreducible algebraic curve defined over a finite field 𝔽_q of characteristic p>2. Assume that the 𝔽_q-automorphism group of 𝒳 admits a subgroup isomorphic to the direct product of two cyclic groups C_m and C_n of orders m and n prime to p, such that both quotient curves 𝒳∕C_n and 𝒳∕C_m are rational. In this paper, we provide a complete classification of such curves as well as a characterization of their full automorphism groups.     

不动点集为RP(4)∪ P(4,2n-1)的对合

设(M,T)是一个光滑闭流形上的对合,不动点集为F=RP(4)UP(4,2n-1),则它的每一个对合(M,T)必协边(RP(4)×RP(4),twist)和(P(4,2n),T')之一.     

On Isometric Actions of SL(n,Z) on Three-Dimensional Visibility Manifolds

Let X be a three-dimensional, visibility manifold. Suppose L(n,Z), n3 acts on X by isometries. Then either the action is trivial or a point in the boundary of X is fixed by the action.     

Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields II

Let k be a separably closed field. Let G be a reductive algebraic k-group. We study Serre’s notion of complete reducibility of subgroups of G over k. In particular, using the recently proved center conjecture of Tits, we show that the centralizer of a k-subgroup H of G is G-completely reducible over k if it is reductive and H is G-completely reducible over k. We show that a regular reductive k-subgroup of G is G-completely reducible over k. We present examples where the number of overgroups of irreducible subgroups and the number of G(k)-conjugacy classes of k-anisotropic unipotent elements are infinite.