Automorphisms of PGL_2(K) over Any Skew Field K

<正> Let K be a skew field and Z be its center.The multiplicative groups of non-zero ele-ments of K and Z are denoted by K~* and Z~* respectively.Denote by GL_2(K) the groupof all 2×2 matrices over K and by PGL_2(K) the factor group of GL_2(K) modulo its ce-nter consisting of all elements of the form γI,where γ∈Z~* and I is the 2×2 identitymatrix.If K is commutative or is of characteristic≠0 the automorphisms of PGL_2(K)are already determined.In the present paper,we'll determine the automorphisms of PGL_2(K)     

Translation ovoids over skew fields

We extend Tits' characterization of the translation ovoids in PG(n, F) to the case that F is a skew field, and give examples. We show that, for n3, a translation ovoid in PG(n, F) is (p, q) transitive for some (every) {p, q}F is communtative and is a hyperquadric.Dedicated to Professor Dr Helmut R. Salzmann on the occasion of his 60th birthday     

Some properties of skew codes over finite fields

After recalling the definition of some codes as modules over skew polynomial rings, whose multiplication is defined by using an endomorphism and a derivation, and some basic facts about them, in the first part of this paper we study some of their main algebraic and geometric properties. Finally, for module skew codes constructed only with an automorphism, we give some BCH type lower bounds for their minimum distance.     

Automorphisms of split Cayley algebras over real closed fields

Research supported in part by NSERC Canada grant A7251.     

On K(3)4 of curves over number fields

In this paper we consider the group K ₄ ⁽³⁾ (F) of a field, in case F is the function field of a smooth geometrically irreducible curve over a number field. We do this using the complexes constructed in [4], together with an auxiliary complex. On the image in K ₄ ⁽³⁾ (F) of those complexes, we derive a formula for the Beilinson regulator, and compute an approximation of the boundary map at the closed points of the curve in the localization sequence. We give a way of finding examples of elliptic curves E with elements in K ₄ ⁽³⁾ (E), and in some cases use computer calculations to check numerically the relation between the regulator and the L-function, as conjectured by Beilinson. Oblatum 10-IX-1995 & 8-I-1996     

Linear liftings of skew symmetric tensor fields of type (1, 2) to Weil bundles

The paper contains a classification of linear liftings of skew symmetric tensor fields of type (1, 2) on n-dimensional manifolds to tensor fields of type (1, 2) on Weil bundles under the condition that n ⩾ 3. It complements author’s paper “Linear liftings of symmetric tensor fields of type (1, 2) to Weil bundles” (Ann. Polon. Math. 92, 2007, pp. 13–27), where similar liftings of symmetric tensor fields were studied. We apply this result to generalize that of author’s paper “Affine liftings of torsion-free connections to Weil bundles” (Colloq. Math. 114, 2009, pp. 1–8) and get a classification of affine liftings of all linear connections to Weil bundles.     

保体上上三角幂等矩阵的诱导映射

假设T_m(D)是体D上所有上三角m×m矩阵的集合.首先分别给出诱导映射和保幂等性的定义.然后为了刻画T_m(D)的保幂等的诱导映射,提出类序列的概念,同时描述类序列的性质.最后,使用矩阵技术和初等方法,借助于分类讨论得到了T_m(D)的保幂等的诱导映射的一般形式并且给出了某些例子,用以解释某些结果之间的关系.     

Orthogonal Groups 𝒪(n) over GF(2) as Automorphisms

Let (V, Q) be a quadratic vector space over a fixed field. Orthogonal group 𝒪(V, Q) is defined as automorphisms on (V, Q). If Q = I, it is 𝒪(V, I) = 𝒪(n). There is a nice result that 𝒪(n) ? Aut(𝔬(n)) over ? or ?, where 𝔬(n) is the Lie algebra of n × n alternating matrices over the field. How about another field The answer is “Yes” if it is GF(2). We show it explicitly with the combinatorial basis ?. This is a verification of Steinberg's main result in 1961, that is, Aut(𝔬(n)) is simple over the square field, with a nonsimple exception Aut(𝔬(5)) ? 𝒪(5) ? 𝔖₆.     

The skew field of rational functions onGL q(n,K)

Samara State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 28, No. 2, pp. 75–77, April–June, 1994.     

k-Involutions of SL(n,k) over fields of characteristic 2

Symmetric k-varieties generalize Riemannian symmetric spaces to reductive groups defined over arbitrary fields. For most perfect fields, it is known that symmetric k-varieties are in one-to-one correspondence with isomorphy classes of k-involutions. Therefore, it is useful to have representatives of each isomorphy class in order to describe the k-varieties. Here we give matrix representatives for each isomorphy class of k-involutions of SL(n,k) in the case that k is any field of characteristic 2; we also describe fixed-point groups of each type of involution.     

Generators for cubic surfaces with two skew lines over finite fields

Let S be a smooth cubic surface defined over a field K. As observed by Segre [5] and Manin [3, 4], there is a secant and tangent process on S that generates new K-rational points from old ones. It is natural to ask for the size of a minimal generating set for S(K). In a recent paper, for fields K with at least 13 elements, Siksek [7] showed that if S contains a skew pair of K-lines, then S(K) can be generated from one point. In this paper we prove the corresponding version of this result for fields K having at least 4 elements, and slightly milder results for # K = 2 or 3.     

Elliptic curves and their torsion subgroups over number fields of type (2, 2, ..., 2)

Suppose thatE: y ² =x(x + M) (x + N) is an elliptic curve, whereM N are rational numbers (#0, ±1), and are relatively prime. LetK be a number field of type (2,...,2) with degree 2′. For arbitrary n, the structure of the torsion subgroup E(K) _tors of theK-rational points (Mordell group) ofE is completely determined here. Explicitly given are the classification, criteria and parameterization, as well as the groups E(K) _tors themselves. The order of E( K)_tors is also proved to be a power of 2 for anyn. Besides, for any elliptic curveE over any number field F, it is shown that E( L)_tors = E( F) _tors holds for almost all extensionsL/F of degree p(a prime number). These results have remarkably developed the recent results by Kwon about torsion subgroups over quadratic fields.     

Eine Charakterisierung der zwischenPS-L (2,K) undPGL (2,K) liegenden Permutationsgruppen

Ohne Zusammenfassung