Elliptic K3 Surfaces with Abelian and Dihedral Groups of Symplectic Automorphisms

We analyze K3 surfaces admitting an elliptic fibration ? and a finite group G of symplectic automorphisms preserving this elliptic fibration. We construct the quotient elliptic fibration ?/G comparing its properties to the ones of ?.

We show that if ? admits an n-torsion section, its quotient by the group of automorphisms induced by this section admits again an n-torsion section, and we describe the coarse moduli space of K3 surfaces with a given finite group contained in the Mordell–Weil group.

Considering automorphisms coming from the base of the fibration, we find the Mordell–Weil lattice of a fibration described by Kloosterman, and we find K3 surfaces with dihedral groups as group of symplectic automorphisms. We prove the isometries between lattices described by the author and Sarti and lattices described by Shioda and by Greiss and Lam.     

Representing Groups by Colourings of Graphs

The topic of this paper is representing groups by edge-coloured graphs. Every edge-coloured graph determines a group of graph automorphisms which preserve the colours of the edges. An edge colouring of a graph G is called a perfect one iff every colour class is a perfect matching in G. We prove that every group H and all of its subgroups can be represented (up to isomorphism) by a group of colour preserving automorphisms related to some perfect colouring of the same graph.     

Symmetry Groups of Coloured Graphs

A perfect colouring Φ of a simple undirected connected graph G is an edge colouring such that each vertex is incident with exactly one edge of each colour. This paper concerns the problem of representing groups by graphs with perfect colourings. We define groups of graph automorphisms, which preserve the structure of the colouring, and characterize these groups up to isomorphism. Our considerations are based on the fact that every perfectly coloured graph is isomorphic to a Schreier coset graph on a group generated by involutions.     

An algorithmic construction of group automorphisms and the quantum Yang-Baxter equation

The structure group G of a non-degenerate symmetric set (X,S) is a Bieberbach and a Garside group. We describe a combinatorial method to compute explicitly a group of automorphisms of G and show this group admits a subgroup that preserves the Garside structure. In some special cases, we could also prove the group of automorphisms found is an outer automorphism group.     

Lie algebras of infinitesimal norm isometries

Given a norm on a finite dimensional vector space V, we may consider the group of all linear automorphisms which preserve it. The Lie algebra of this group is a Lie subalgebra of the endomorphism algebra of V having two properties: (1) it is the Lie algebra of a compact subgroup, and (2) it is “saturated” in a sence made precise below. We show that any Lie subalgebra satisfying these conditions is the Lie algebra of the group of linear automorphisms preserving some norm. There is an appendix on elementary Lie group theory.     

The centralizer of a set of group automorphisms

Let A be a set of automorphisms of a finite group G. The structure of the near-ring C(A) of identity preserving functions which commute with every element of A is investigated.     

Complete nondegenerate locally standard CR manifolds

Abstract. Let M be a complete nondegenerate locally standard CR manifold. We show that a necessary and sufficient condition for M to be compact is that the Lie algebra of its infinitesimal CR automorphisms is semisimple. In general we realize M as a Mostow fibration over a compact CR manifoldB whose universal covering is a Cartesian product of Hermitian symmetric spaces and compact nondegenerate standard CR manifolds. Received: 22 July 1998 / Published online: 8 May 2000     

Fibrations associated to smooth quotients of abelian varieties

Geometriae Dedicata - Let A be a complex abelian variety and G a finite group of automorphisms of A fixing the origin such that A/G is smooth. The quotient A/G can be seen as a fibration over an...     

一类中心循环的有限p-群的自同构群的研究

本文研究了一类中心循环的有限p-群G的自同构群.利用在G的导群上作用平凡的自同构以及环上的辛群和正交群,确定了G的自同构群的结构,这推广了Bornand的相应结果.     

Embeddings of SL(2; ?) into the cremona group

Geometric and dynamic properties of embeddings of SL(2; ℤ) into the Cremona group are studied. Infinitely many nonconjugate embeddings that preserve the type (i.e., that send elliptic, parabolic and hyperbolic elements onto elements of the same type) are provided. The existence of infinitely many nonconjugate elliptic, parabolic and hyperbolic embeddings is also shown. In particular, a group G of automorphisms of a smooth surface S obtained by blowing up 10 points of the complex projective plane is given. The group G is isomorphic to SL(2; ℤ), preserves an elliptic curve and all its elements of infinite order are hyperbolic.     

X-Inner Automorphisms,Automorphisms, and Isomorphisms of Skew Polynomial Rings

Abstract

In this paper, we examine the X-inner automorphisms, automorphisms, and isomorphisms of skew polynomial rings of the form K[x][y;δ], where K is a field of characteristic 0 and δ is a derivation of K [x] such that x ^δ is a polynomial of degree ≥ 1. We 1.  determine the group of X-inner automorphisms of K [x][ y;δ],

2.  analyze the structure of the group of automorphisms of K [x][ y;δ], and

3.  examine the isomorphism classes of skew polynomial rings of the form K [x][ y;δ].

We also provide several examples which indicate the importance of the base field in computing the X-inner automorphisms, automorphisms, and isomorphism classes of K [x][ y;δ].     

Classification of sequentially weakly continuous $JB^*$-triples

Let D be the open unit ball of a -triple A and let Aut(D) be the group of all biholomorphic automorphisms of D. It is shown that every element of Aut(D) is sequentially weakly continuous if and only if every primitive ideal of A is a maximal closed ideal and is a type I -triple without infinite-spin part. Implications for general structure theory are explored. In particular, it is deduced that every -triple A contains a smallest ideal J for which the sequentially weakly continuous biholomorphic automorphisms of the open unit ball of A/J are all linear. Received August 27, 1998; in final form February 10, 1999     

Weakly power automorphisms of groups

An automorphism α of a group G is called a weakly power automorphism if it maps every non-periodic subgroup of G onto itself. The aim of this paper is to investigate the behavior of weakly power automorphisms. In particular, among other results, it is proved that all weakly power automorphisms of a soluble non-periodic group G of derived length at most 3 are power automorphisms, i.e. they fix all subgroups of G. This result is best possible, as there exists a soluble non-periodic group of derived length 4 admitting a weakly power automorphism, which is not a power automorphism.     

On the weak isometries of the Boolean cube

The p-weak isometries of the Boolean cube are considered, i.e., the mappings of the cube onto itself which preserve a fixed distance p. Some necessary and sufficient conditions are obtained for coincidence of the group of p-weak isometries with the group of standard isometries (automorphisms of the cube).     

Suzuki群Sz(q)和线性空间的自同构群

本文研究了线性空间的几乎单的线传递自同构群.利用有限线性空间上线传递自同构群的经典结论,以及Suzuki群Sz(q)的性质,获得了线性空间上线传递且点本原的自同构群的基柱不是Sz(q)的结果,推广了关于线传递性空间的已有结果.     

The envelope of holomorphy of a model third-degree surface and the rigidity phenomenon

The structures of the graded Lie algebra aut Q infinitesimal automorphisms of a cubic (a model surface in ?^N) and the corresponding group Aut Q of its holomorphic automorphisms are studied. It is proved that for any nondegenerate cubic, the positively graded components of the algebra aut Q are trivial and, as a consequence, Aut Q has no subgroups consisting of nonlinear automorphisms of the cubic that preserve the origin (the so-called rigidity phenomenon). In the course of the proof, the envelope of holomorphy for a nondegenerate cubic is constructed and shown to be a cylinder with respect to the cubic variable whose base is a Siegel domain of the second kind.     

On Generators of the Tame Automorphism Group of Free Metabelian Lie Algebras

We prove that the tame automorphism group TAut(M _n ) of a free metabelian Lie algebra M _n in n variables over a field k is generated by a single nonlinear automorphism modulo all linear automorphisms if n ≥ 4 except the case when n = 4 and char(k) ≠ 3. If char(k) = 3, then TAut(M ₄) is generated by two automorphisms modulo all linear automorphisms. We also prove that the tame automorphism group TAut(M ₃) cannot be generated by any finite number of automorphisms modulo all linear automorphisms.     

Power automorphisms of finitep-groups

For a finite groupG letA(G) denote the group of power automorphisms, i.e. automorphisms normalizing every subgroup ofG. IfG is ap-group of class at mostp, the structure ofA (G) is shown to be rather restricted, generalizing a result of Cooper ([2]). The existence of nontrivial power automorphisms, however, seems to impose restrictions on thep-groupG itself. It is proved that the nilpotence class of a metabelianp-group of exponentp ² possessing a nontrival power automorphism is bounded by a function ofp. The “nicer” the automorphism—the lower the bound for the class. Therefore a “type” for power automorphisms is introduced. Several examples ofp-groups having large power automorphism groups are given.