The t-designs with prescribed automorphism group,new simple 6-designs

We introduce an algorithm for the construction of a complete system of representatives of t-designs with given parameters t ? (v, k, λ) and prescribed full automorphism group A. It is based on the following observation published by Kramer and Mesner in 1976: The t ? (v, k, λ) designs admitting automorphism group A are exactly the 0-1-solutions $ \overrightarrow x $ of the following system of linear equations M are incidence matrices, which we compute by means of double cosets. Representing the set of all solutions of the above system of equations implicitly by a graph gives us the possibility either to extract the solutions explicitly or to compute their precise numbers, which often are very big. We use the lattice of overgroups of A in the full symmetric group S_v for the construction or enumeration of the isomorphism types of the t-designs with full automorphism group A from these solutions. To the best of our knowledge our approach for the first time allows one to compute the precise number of isomorphism types or even these designs themselves for substantial numbers. We determined the (number of) isomorphism types for many known parameter sets and found new simple 6-designs with parameters and full automorphism group PΓL₂(27). We constructed all isomorphism types of these designs; their precise numbers are 3,367,21 743,38 277, respectively. © 1993 John Wiley & Sons, Inc.     

A simple example of two p-groups with the same automorphism group

It is an open problem whether there exist two different p-groups with the same automorphism group. In this paper, we give a positive answer to this problem by constructing a simple example. Received: 30 September 2008     

The automorphism group of the augmented equiaffine group

The automorphism group of the group generated by all the affine reflections in a Desarguesian plane is isomorphic to the full collineation group of the plane.The author wishes to express his thanks to J. Wilker of the University of Toronto for suggesting a simplification and to him as well as C. Fisher of the University of Regina for their encouragement.     

The automorphism group of the tetrablock

The tetrablock is shown to be inhomogeneous and its automorphismgroup is determined. A type of Schwarz lemma for the tetrablockis proved. The action of the automorphism group is describedin terms of a certain natural foliation by complex geodesicdiscs.     

On the automorphism groups of symmetric graphs admitting an almost simple group

This paper investigates the automorphism group of a connected and undirected G-symmetric graph Γ where G is an almost simple group with socle T. First we prove that, for an arbitrary subgroup M of containing G, either T is normal in M or T is a subgroup of the alternating group A_k of degree . Then we describe the structure of the full automorphism group of G-locally primitive graphs of valency d, where d≤20 or is a prime. Finally, as one of the applications of our results, we determine the structure of the automorphism group for cubic symmetric graph Γ admitting a finite almost simple group.     

The automorphism group of a coded system

We give a general construction of coded systems with an automorphism group isomorphic to where is any preassigned group which has a ``continuous block presentation' (the isomorphism will map the shift to . Several applications are given. In particular, we obtain automorphism groups of coded systems which are abelian, which are finitely generated and one which contains . We show that any group which occurs as a subgroup of the automorphism group of some subshift with periodic points dense already occurs for some synchronized system.

     

圈的自同结构群的阶

n个顶点的圈Cn的自同结构群的阶是2n.     

The automorphism group of a circle geometry

The following is proved: Theorem. Let n be an integer, n ? 2. Let F be a field with at least Max(n, 3) distinct elements and let K be a separable extension field of dimension n over F. Then the group of all automorphisms of the n-dimensional circle geometry CG(F, K) is the group induced on the 1-dimensional vector subspaces over K of the underlying 2-dimensional vector space V over K by the semi-linear permutations of V over K for which the associated automorphisms of K induce automorphisms of F.With the hope of broadening the interest in the foundations of geometry, the author attempts to give anough detail so that, without consulting the literature, any attentive algebraist can understand clearly both the content of the theorem and the pattern of its proof and can identify easily any gaps in his knowledge which might prevent him from viewing this paper as a contribution to geometry. Adequate references are given.     

An automorphism with simple,continuous spectrum not having the group property

An example is constructed of an automorphism with simple and continuous spectrum which does not have the group property.Translated from Matematicheskie Zametki, Vol. 5, No. 3, pp. 323–326, March, 1969.