Divisible designs with dual translation group

Many different divisible designs are already known. Some of them possess remarkable automorphism groups, so called dual translation groups. The existence of such an automorphism group enables us to characterize its associated divisible design as being isomorphic to a substructure of a finite affine space.      

Characterizations by Automorphism Groups of some Rank 3 Buildings – I. Some Properties of Half Strongly-Transitive Triangle Buildings

In a sequence of papers, we will show that the existence of a (half) strongly-transitive automorphism group acting on a locally finite triangle building forces to be one of the examples arising from PSL₃(K) for a locally finite local skewfield K. Furthermore, we introduce some Moufang-like conditions in affine buildings of rank 3, and characterize those examples arising from algebraic, classical or mixed type groups over a local field. In particular, we characterize the p-adic-like affine rank 3 buildings by a certain p-adic Moufang condition, and show that such a condition has zero probability to survive in hyperbolic rank 3 buildings. This shows that a construction of hyperbolic buildings as analogues of p-adic affine buildings is very unlikely to exist.     

Symmetries of the Partial Order of Traces

This paper deals with the automorphism group of the partial order of finite traces. We show that any group can arise as such an automorphism group if we allow arbitrary large dependence alphabets. Restricting to finite dependence alphabets, the automorphism groups are profinite and possess only finitely many simple decomposition factors. Finally, we show that the partial order associated with the Rado graph as dependence alphabet does not give rise to a homogeneous domain thereby answering an open question from Boldi, P., Cardone, F. and Sabadini, N. (1993).     

Suzuki群与Steiner 4设计

本文研究了非平凡Steiner 4设计的自同构群是旗传递的情形.利用有限2传递置换群的分类,得到了旗传递非平凡Steiner 4设计的自同构群的基柱不是Suzuki群.     

Periodicities in Cluster Algebras and Cluster Automorphism Groups

We study the relations between two groups related to cluster automorphism groups which are defined by Assem,Schiffler and Shamchenko.We establish the relation-ships among (strict) direct cluster automorphism groups and those groups consisting of periodicities of labeled seeds and exchange matrices,respectively,in the language of short exact sequences.As an application,we characterize automorphism-finite cluster algebras in the cases of bipartite seeds or finite mutation type.Finally,we study the relation between the group Aut(A) for a cluster algebra A and the group AutMn(S) for a mutation group Mn and a labeled mutation class S,and we give a negative answer via counter-examples to King and Pressland's problem.     

Sporadic groups and automorphisms of linear spaces

This article is a contribution to the study of the automorphism groups of finite linear spaces. In particular we look at almost simple groups and prove the following theorem: Let G be an almost simple group and let 𝒮 be a finite linear space on which G acts as a line‐transitive automorphism group. Then the socle of G is not a sporadic group. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 353–362, 2000     

一类不能作为自同构群的奇阶群

本文考虑如下问题：怎样的有限群可以作为另一个有限群的全自同构群？我们首先证明，若有限群Ｋ有一个正规Ｓｙｌｏｗｐ－子群使得｜Ｋ：Ｚ（Ｋ）｜ｐ＝ｐ２，那么Ｋ有２阶自同构．利用这个结果，我们证明了，若奇阶群Ｇ具有阶Ｐｓｍ（１≤ｓ≤３），ｐ为｜Ｇ｜的最小素因子，ｐｍ，ｍ无立方因子，则Ｇ不可能作为全自同构群．     

Kollineationen von Translationsstrukturen

Translationstructures are generalized affine spaces. They can be described algebraically by partitions of groups. For desarguesian affine spaces the group is a vectorspace and the partition is the set of all onedimensional subspaces. In this case each collineation fixing 0 is a regular semilinear mapping, i.e. an automorphism of the vectorspace. In the general case it is a mapping called equivalence. Each equivalence of a partition is an automorphism iff the set of translations of the group is a normal subgroup of the collineationgroup. The translations form a normal subgroup, if the group is finite or abelian. We prove some theorems for the infinite non abelian case.     

On Some Automorphisms of Orthogonal Groups in Odd Characteristic

In this paper, it is proved that the simple orthogonal groups O _2n+1(q) and O _2n ^± (q) (where q is odd) cannot be automorphism groups of finite left distributive quasigroups. This is a particular case of the conjecture stating that the automorphism group of a left distributive quasigroup is solvable. To complete the proof of the conjecture, one must test all finite groups.     

Lie group structures on symmetry groups of principal bundles

In this paper we describe how one can obtain Lie group structures on the group of (vertical) bundle automorphisms for a locally convex principal bundle P over the compact manifold M. This is done by first considering Lie group structures on the group of vertical bundle automorphisms Gau(P). Then the full automorphism group Aut(P) is considered as an extension of the open subgroup DiffP(M) of diffeomorphisms of M preserving the equivalence class of P under pull-backs, by the gauge group Gau(P). We derive explicit conditions for the extensions of these Lie group structures, show the smoothness of some natural actions and relate our results to affine Kac-Moody algebras and groups.     

Duality in stable planes and related closure and kernel operations

We compare two constructions that dualize a stable plane in some sense, namely the dual plane and the opposite plane. Applying both constructions one after another we obtain a closure or kernel operation, depending on the order of execution.We examine the effect of these constructions on the automorphism group and apply our results in order to compute the automorphism groups of the complex cylinder plane, the complex united cylinder plane, and their duals. Beside the complex projective, affine, and punctured projective plane these planes are in fact the most homogeneous four-dimensional stable planes, as will be shown elsewhere [1].Supported by Studienstiftung des deutschen Volkes.     

Direct Products with Abelian Automorphism Groups

The article considers when the direct product of two finite groups has an Abelian automorphism group.     

Realizing finite edge‐transitive orientable maps

J.E. Graver and M.E. Watkins, Memoirs Am. Math. Soc. 126 (601) ( 5 ) established that the automorphism group of an edge‐transitive, locally finite map manifests one of exactly 14 algebraically consistent combinations (called types) of the kinds of stabilizers of its edges, its vertices, its faces, and its Petrie walks. Exactly eight of these types are realized by infinite, locally finite maps in the plane. H.S.M. Coxeter (Regular Polytopes, 2nd ed., McMillan, New York, 1963) had previously observed that the nine finite edge‐transitive planar maps realize three of the eight planar types. In the present work, we show that for each of the 14 types and each integer n ≥ 11 such that n ≡ 3,11 (mod 12), there exist finite, orientable, edge‐transitive maps whose various stabilizers conform to the given type and whose automorphism groups are (abstractly) isomorphic to the symmetric group Sym(n). Exactly seven of these types (not a subset of the planar eight) are shown to admit infinite families of finite, edge‐transitive maps on the torus, and their automorphism groups are determined explicitly. Thus all finite, edge‐transitive toroidal maps are classified according to this schema. Finally, it is shown that exactly one of the 14 types can be realized as an abelian group of an edge‐transitive map, namely, as ?_n × ?₂ where n ≡ 2 (mod 4). © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 1–34, 2001     

Automorphisms and Isomorphisms of Symmetric and Affine Designs

Given a finite group G, for all sufficiently large d and for each q > 3 there are symmetric designs and affine designs having the same parameters as PG(d, q) and AG(d, q), respectively, and having full automorphism group isomorphic to G.     

Curran猜想的解答

本文确定了能作为有限群自同构群的Ｐ－群的最佳下最，彻底解决了Ｃｕｒｒａｎ在１９８９年提出的关于此最佳下界的三个猜想。     

具有4pq阶自同构群的有限群

本文讨论了自同构群阶为4pq(p,q为不同奇素数)的有限群,得出了它们的构造.     

Infinite dimensional general linear groups have finite width

A group is said to have finite width whenever it has finite width with respect to each inverse-closed generating set. Bergman showed [1] that infinite symmetric groups have finite width and asked whether the automorphism groups of several classical structures have finite width, mentioning in particular infinite dimensional general linear groups over fields. In this article we prove that infinite dimensional general linear groups over arbitrary division rings have finite width. We consider the problem of finite width for other infinite dimensional classical groups.     

Generalized quadrangles admitting a sharply transitive Heisenberg group

All known finite generalized quadrangles that admit an automorphism group acting sharply transitively on their point set arise by Payne derivation from thick elation generalized quadrangles of order s with a regular point. In these examples only two groups occur: elementary abelian groups of even order and odd order Heisenberg groups of dimension 3. In [2] the authors determined all generalized quadrangles admitting an abelian group with a sharply transitive point action. Here, we classify thick finite generalized quadrangles admitting an odd order Heisenberg group of dimension 3 acting sharply transitively on the points. In fact our more general result comes close to a complete solution of classifying odd order Singer p-groups.