Gyrosemidirect Product Structure of Bounded Symmetric Domains

A binary operation ⊕ is defined in any bounded symmetric domain D turning it into a groupoid with relaxed associative and commutative laws, called a gyrogroup. It is shown that the group Aut(D) of all holomorphic automorphisms of D has a gyrosemidirect product structure, a structure that generalizes the semidirect product one. More specifically, the group Aut(D) turns out to be the gyrosemidirect product of the (nongroup) gyrogroup (D, ⊕) and the isotropic group K of D.     

finite groups with schmidt group as automorphism group

This paper continues the work of D.MacHale,D.Flannery(Proc.R.Ir.Acad.81A,209—215;83A,189—196)and the author(Proc.R.Ir.Acad,90A,57—62;J.Southwest China Normal University 15,No.1,21.—28)on the topic on“Finite groupswith given Automorphism group”.The following result is proved:Let G be a finite group with Aut G a Schmidt group.Then G is isomorphic toS_3 or Klain 4-group.,or D such that Aut D=Inn D.D is aSchmidt group of order 2~(?)p.S_2(∈Syl_2D)is a normal and special group exoept asupersperspecial group without commutative generators.     

Suzuki群Sz(q)与2-(v,k,1)区组设计

§ 1　IntroductionA2 -( v,k,1 ) design D=( Ω,B) is a system consisting of a finite setΩ ofv points anda collection Bofk-subsets ofΩ ,called blocks,such thatany 2 -subsetofΩ is contained inexactly one block.We shall always assume that2 相似文献   

Explicit formula of holomorphic automorphism group on complex homogeneous bounded domains

We known that the maximal connected holomorphic automorphism group Aut (D)(0) is a semi-direct product of the triangle group T(D) and the maximal connected isotropic subgroup Iso(D)(0) of a fixed point in the complex homogeneous bounded domain D and any complex homogeneous bounded domain is holomorphic isomorphic to a normal Siegel domain D(VN,F). In this paper, we give the explicit formula of any holomorphic automorphism in T(D(VN, F)) and Iso(D(VN,F))(0), where G(0) is the unit connected component of the Lie group G.     

Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions

This paper gives $n$-dimensional analogues of the Apollonian circle packings in Parts I and II. Those papers considered circle packings described in terms of their Descartes configurations, which are sets of four mutually touching circles. They studied packings that had integrality properties in terms of the curvatures and centers of the circles. Here we consider collections of $n$-dimensional Descartes configurations, which consist of $n+2$ mutually touching spheres. We work in the space $M_D^n$ of all $n$-dimensional oriented Descartes configurations parametrized in a coordinate system, augmented curvature-center coordinates, as those $(n+2) \times (n+2)$ real matrices $W$ with $W^T Q_{D,n} W = Q_{W,n}$ where $Q_{D,n} = x_1^2 + \cdots + x_{n+2}^2 - ({1}/{n})(x_1 +\cdots + x_{n+2})^2$ is the $n$-dimensional Descartes quadratic form, $Q_{W,n} = -8x_1x_2 + 2x_3^2 + \cdots + 2x_{n+2}^2$, and $\bQ_{D,n}$ and $\bQ_{W,n}$ are their corresponding symmetric matrices. On the parameter space $M_D^n$ of augmented curvature-center matrices, the group ${\it Aut}(Q_{D,n})$ acts on the left and ${\it Aut}(Q_{W,n})$ acts on the right. Both these groups are isomorphic to the $(n+2)$-dimensional Lorentz group $O(n+1,1)$, and give two different "geometric" actions. The right action of ${\it Aut}(Q_{W,n})$ (essentially) corresponds to Mobius transformations acting on the underlying Euclidean space $\rr^n$ while the left action of ${\it Aut}(Q_{D,n})$ is defined only on the parameter space $M_D^n$. We introduce $n$-dimensional analogues of the Apollonian group, the dual Apollonian group and the super-Apollonian group. These are finitely generated groups in ${\it Aut}(Q_{D,n})$, with the following integrality properties: the dual Apollonian group consists of integral matrices in all dimensions, while the other two consist of rational matrices, with denominators having prime divisors drawn from a finite set $S$ depending on the dimension. We show that the Apollonian group and the dual Apollonian group are finitely presented, and are Coxeter groups. We define an Apollonian cluster ensemble to be any orbit under the Apollonian group, with similar notions for the other two groups. We determine in which dimensions there exist rational Apollonian cluster ensembles (all curvatures are rational) and strongly rational Apollonian sphere ensembles (all augmented curvature-center coordinates are rational).     

Groups of homeomorphisms with manageable automorphism groups

This paper explores the automorphisms of certain groups of homeomorphisms pf the real line ?, namely those which are o-2-transitive and contain positive elements of bounded support. If G is such a group, every automorphism of G is induced (via conjugation) by some elenent of the group M(?) of all homeomorphisms of ?, and we may presume that Aut(G) ≤ ?. Thus each automorphism either preserves or reverses order, and this is the key to the proof. This generalizes results of James Whittaker [l9] so as to encompass the group D of diffeomorphisms (C¹not required) and the group PL of piecewise linear homeomorphisms, for which it is further shown that all auto­morphisms are inner. Other aspects of these two groups were dealt with in [15], for which the present paper is a companion. Similar results are obtained for more general ordered sets S, but with Aut (G) ≤ M([Sbar]) [Sbar]the Dedekind completion of S. In many cases, however, this can be improved to yield Aut(G) ≤ M(S).     

On the Homeomorphism Groups of Cantor's Discontinuum and the Spaces of Rational and Irrational Numbers

The paper shows that the homeomorphism groups of, respectively,Cantor's discontinuum, the rationals and the irrationals haveuncountable cofinality. It is well known that the homeomorphismgroup of Cantor's discontinuum is isomorphic to the automorphismgroup Aut B of the countable, atomless boolean algebra B. Soalso Aut B has uncountable cofinality, which answers a questionposed earlier by the first author and H. D. Macpherson. Thecofinality of a group G is the cardinality of the length ofa shortest chain of proper subgroups terminating at G. 2000Mathematics Subject Classification 20B22, 20E15.     

素特征域上广义Witt李超代数的自同构群

设W是素特征域上无限维或有限维广义Witt李超代数.本文利用W的自然滤过不变性和W的底代数的不变维数性质,证明了W的自同构群AutW同构于W的底代数的容许自同构群,还证明了在此群同构之下,AutW的标准正规列恰好对应W的底代数的容许自同构群的标准正规列,并给出AutW若干较为细致的性质.     

本原(J1,2)-弧传递图的分类

In the author's Ph. D thesis, a non-quasiprimitive graph admitting a quasiprimitive automorphism group isomorphic to J1 was constructed,where J1 is Janko simple group of order 175560.Is this the only one for J1?In this paper all primitive(J1,2)-arc transitive graphs Γ are given and that AutΓ≌J1 is proved.     

Apollonian Circle Packings: Geometry and Group Theory I. The Apollonian Group

Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We observe that there exist Apollonian packings which have strong integrality properties, in which all circles in the packing have integer curvatures and rational centers such that (curvature) $\times$ (center) is an integer vector. This series of papers explain such properties. A Descartes configuration is a set of four mutually tangent circles with disjoint interiors. An Apollonian circle packing can be described in terms of the Descartes configuration it contains. We describe the space of all ordered, oriented Descartes configurations using a coordinate system $M_ D$ consisting of those $4 \times 4$ real matrices $W$ with $W^T Q_{D} \bW = Q_{W}$ where $Q_D$ is the matrix of the Descartes quadratic form $Q_D= x_1^2 + x_2^2+ x_3^2 + x_4^2 - \frac{1}{2}(x_1 +x_2 +x_3 + x_4)^2$ and $Q_W$ of the quadratic form $Q_W = -8x_1x_2 + 2x_3^2 + 2x_4^2$. On the parameter space $M_ D$ the group $\mathop{\it Aut}(Q_D)$ acts on the left, and $\mathop{\it Aut}(Q_W)$ acts on the right, giving two different "geometric" actions. Both these groups are isomorphic to the Lorentz group $O(3, 1)$. The right action of $\mathop{\it Aut}(Q_W)$ (essentially) corresponds to Mobius transformations acting on the underlying Euclidean space $\rr^2$ while the left action of $\mathop{\it Aut}(Q_D)$ is defined only on the parameter space. We observe that the Descartes configurations in each Apollonian packing form an orbit of a single Descartes configuration under a certain finitely generated discrete subgroup of $\mathop{\it Aut}(Q_D)$, which we call the Apollonian group. This group consists of $4 \times 4$ integer matrices, and its integrality properties lead to the integrality properties observed in some Apollonian circle packings. We introduce two more related finitely generated groups in $\mathop{\it Aut}(Q_D)$, the dual Apollonian group produced from the Apollonian group by a "duality" conjugation, and the super-Apollonian group which is the group generated by the Apollonian and dual Apollonian groups together. These groups also consist of integer $4 \times 4$ matrices. We show these groups are hyperbolic Coxeter groups.     

Heptavalent Symmetric Graphs with Certain Conditions

A graph Γ is said to be symmetric if its automorphism group Aut(Γ)acts transitively on the arc set of Γ.We show that if Γ is a finite connected heptavalent symmetric graph with solvable stabilizer admitting a vertex-transitive non-abelian simple group G of automorphisms,then either G is normal in Aut(Γ),or Aut(Γ)contains a non-abelian simple normal subgroup T such that G≤T and(G,T)is explicitly given as one of 11 possible exceptional pairs of non-abelian simple groups.If G is arc-transitive,then G is always normal in Aut(r),and if G is regular on the vertices of Γ,then the number of possible exceptional pairs(G,T)is reduced to 5.     

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     

某些Lie单群的传递性

设D为有限线性空间，且T G Aut（T），其中T是非交换单群，并且同构于^2B2（g），Cn（g）（n≥3），^3D4（g），E7（q），E8（q），F4（q），^2F4（q），G2（q），^2G2（q）。假设D不是射影平面，G线传递作用在D上，则T点传递。     

Some Properties of Aut（X） and the Subgroup Aut∑（X）

Let Aut. （X） denote the group of homotopy classes of self-homotopy equivalences of X, which induce identity automorphisms of homology group. We describe a decomposition of Aut. （X1 V…VXn） as a product of its simpler subgroups. We consider the subgroup Aut∑（X） of all self homotopy classes α of X such that ∑α=1∑X： ∑X → ∑X, and also give some properties of Aut∑（X）.     

All Groups are Outer Automorphism Groups of Simple Groups

It is shown that each group is the outer automorphism groupof a simple group. Surprisingly, the proof is mainly based onthe theory of ordered or relational structures and their symmetrygroups. By a recent result of Droste and Shelah, any group isthe outer automorphism group Out (Aut T) of the automorphismgroup Aut T of a doubly homogeneous chain (T, ). However, AutT is never simple. Following recent investigations on automorphismgroups of circles, it is possible to turn (T, ) into a circleC such that Out (Aut T) Out (Aut C). The unavoidable normalsubgroups in Aut T evaporate in Aut C, which is now simple,and the result follows.     

Compactness conditions for groups of automorphisms of topological groups

It is proved that if G is a compact, totally disconnected Abelian group and Aut G is its group of topological automorphisms (with the natural topology), then the following conditions are equivalent: (a) Aut G is compact; (b) Aut G is locally compact; (c) Aut G has small invariant neighborhoods of the identity; (d) Aut G is an -group; (e) the factor group of Aut G by its center is compact; (f) the closure of the commutator subgroup of Aut G is compact; (g) , where F_p is a finite p-group, Z_p is the additive group of p-adic integers, and n_p < .Translated from Matematicheskie Zametki, Vol. 19, No. 5, pp. 735–743, May, 1976.In conclusion, the author thanks V. P. Platonov for his constant attention to this paper.     

拓扑和的自同伦等价群

如果拓扑空间X,Y的拓扑和X∨Y的自同伦等价可以对角化,则X∨Y的自同伦等价群Aut(X∨Y)可表示为它的两个子群Autx(X∨Y)与AutY(X∨y)的乘积.     

Large conjugacy classes,projective Fraïssé limits and the pseudo-arc

We show that the automorphism group, Aut(?), of a projective Fraïssé limit ?, whose natural quotient is the pseudo-arc, has a comeager conjugacy class. This generalizes an unpublished result of Izhar Oppenheim that Aut(?) (and consequently, the group of all homeomorphisms of the pseudo-arc) has a dense conjugacy class. We also present a simple proof of the result of Oppenheim.     

Steinberg triality groups acting on 2 - (v, k, 1) designs

A 2 - (v,k,1) design D = (P, B) is a system consisting of a finite set P of v points and a collection B of k-subsets of P, called blocks, such that each 2-subset of P is contained in precisely one block. Let G be an automorphism group of a 2- (v,k,1) design. Delandtsheer proved that if G is block-primitive and D is not a projective plane, then G is almost simple, that is, T ≤ G ≤ Aut(T), where T is a non-abelian simple group. In this paper, we prove that T is not isomorphic to 3D4(q). This paper is part of a project to classify groups and designs where the group acts primitively on the blocks of the design.