Every group is the automorphism group of a rank-3 matroid

Every group is the automorphism group of a rank-3 extension of a rank-3 Dowling geometry.Partially supported by The George Washington University UFF grant.Partially supported by the National Security Agency under grant MDA904-91-H-0030.     

Every finite group is the group of self-homotopy equivalences of an elliptic space

We prove that every finite group G can be realized as the group of self-homotopy equivalences of infinitely many elliptic spaces X. To construct those spaces we introduce a new technique which leads, for example, to the existence of infinitely many inflexible manifolds. Further applications to representation theory will appear in a separate paper.     

Every group has a terminating transfinite automorphism tower

Iteratively taking the automorphism group of any group leads, transfinitely, to a fixed point.

     

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.     

On automorphism groups of some finite groups

We show that if n > 1 is odd and has no divisor p4 for any prime p, then there is no finite group G such that│Aut(G)│ = n.     

The automorphism groups of the congruence group and of some of its subgroups in finite Euclidean planes

In the affine plane over a Galois field GF(q), q ; 3(4), q = p, of congruence transformations, of motions and of the generation of all point reflections respectively. Then we determine the groups AutC, AutM, AutM and obtain the following results: 1. Aut C is isomorphic to the product of the augmented group of similarities (generated by similarities, quasi reflections, quasi rotations 2) and the group of collineations which are induced by the automorphism of GF(q) operating on the coordinates. 2. AutM– AutC. 3. AutM– group of affinities of the affine space of dimension 2 over the prime field. 4. Moreover for any desarguesian affine plane Aut Dil (Dil = group of dilatations) is isomorphic to (the full collineation group).Lecture delivered at the Haifa Geometry conference 1983In my lecture I called these transformations semi-. To avoid confusion I follow here a suggestion of E. Schröder.     

On the size of the symmetry group of a perfect code

It is shown that for every nonlinear perfect code C of length n and rank r with n−log(n+1)+1≤r≤n−1, where denotes the group of symmetries of C. This bound considerably improves a bound of Malyugin.     

Every compact group arises as the outer automorphism group of a II1 factor

We show that any compact group can be realized as the outer automorphism group of a factor of type II₁. This has been proved in the abelian case by Ioana, Peterson and Popa [A. Ioana, J. Peterson, S. Popa, Amalgamated free products of w-rigid factors and calculation of their symmetry group, math.OA/0505589, Acta Math., in press] applying Popa's deformation/rigidity techniques to amalgamated free product von Neumann algebras. Our methods are a generalization of theirs.