A Steiner 5-Design on 36 Points

Up to now, all known Steiner 5-designs are on q + 1 points where q 3 (mod 4) is a prime power and the design is admitting PSL(2, q) as a group of automorphisms. In this article we present a 5-(36,6,1) design admitting PGL(2, 17) × C ₂ as a group of automorphisms. The design is unique with this automorphism group and even for the commutator group PSL(2, 17) × Id ₂ of this automorphism group there exists no further design with these parameters. We present the incidence matrix of t-orbits and block orbits.     

Modified generalized Hadamard matrices and constructions for transversal designs

It is well known that there exists a transversal design TD_λ[k; u] admitting a class regular automorphism group U if and only if there exists a generalized Hadamard matrix GH(u, λ) over U. Note that in this case the resulting transversal design is symmetric by Jungnickel’s result. In this article we define a modified generalized Hadamard matrix and show that transversal designs which are not necessarily symmetric can be constructed from these under a modified condition similar to class regularity even if it admits no class regular automorphism group.     

Automorphisms of Chevalley groups of types <Emphasis Type="Italic">B</Emphasis><Subscript>2</Subscript> and <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript> over local rings

In the paper, we prove that every automorphism of any adjoint Chevalley group of type B ₂ or G ₂ is standard, i.e., it is a composition of an “inner” automorphism, a ring automorphism, and a central automorphism. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 4, pp. 3–29, 2007.     

Automorphisms and Derivations of Certain Solvable Lie Algebras Over Commutative Rings

Let L be a finite-dimensional complex simple Lie algebra, L _? be the ?-span of a Chevalley basis of L, and L _R  = R ?_? L _? be a Chevalley algebra of type L over a commutative ring R with identity. Let ?(R) be the solvable subalgebra of L _R spanned by the basis elements of the maximal toral subalgebra and the root vectors associated with positive roots. In this article, we prove that under some conditions for R, any automorphism of ?(R) is uniquely decomposed as a product of a graph automorphism, a diagonal automorphism and an inner automorphism, and any derivation of ?(R) is uniquely decomposed as a sum of an inner derivation induced by root vectors and a diagonal derivation. Correspondingly, the automorphism group and the derivation algebra of ?(R) are determined.     

Resolutions of PG(5, 2) with point‐cyclic automorphism group

A t‐(υ, k, λ) design is a set of υ points together with a collection of its k‐subsets called blocks so that all subsets of t points are contained in exactly λ blocks. The d‐dimensional projective geometry over GF(q), PG(d, q), is a 2‐(q^d + q^d−1 + … + q + 1, q + 1, 1) design when we take its points as the points of the design and its lines as the blocks of the design. A 2‐(υ, k, 1) design is said to be resolvable if the blocks can be partitioned as ℛ = {R₁, R₂, …, R_s}, where s = (υ − 1)/(k−1) and each R_i consists of υ/k disjoint blocks. If a resolvable design has an automorphism σ which acts as a cycle of length υ on the points and ℛ^σ = ℛ, then the design is said to be point‐cyclically resolvable. The design associated with PG(5, 2) is known to be resolvable and in this paper, it is shown to be point‐cyclically resolvable by enumerating all inequivalent resolutions which are invariant under a cyclic automorphism group G = 〈σ〉 where σ is a cycle of length 63. These resolutions are the only resolutions which admit a point‐transitive automorphism group. Furthermore, some necessary conditions for the point‐cyclic resolvability of 2‐(υ, k, 1) designs are also given. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 2–14, 2000     

Automorphism groups of some algebras

The automorphism groups of algebras are found in many papers. Using auto-invariance, we find the automorphism groups of the Laurent extension of the polynomial ring and the quantum n-plane (respectively, twisting polynomial ring) in this work. As an application of the results of this work, we can find the automorphism group of a twisting algebra. We define a generalized Weyl algebra and show that the generalized Weyl algebra is simple. We also find the automorphism group of a generalized Weyl algebra. We show that the generalized Weyl algebra A _m,m+n is the universal enveloping algebra of the generalized Witt algebra W(m,m + n). This work was supported by 2007 Research fund of Hanyang University     

Automorphisms of elementary adjoint Chevalley groups of types Al, Dl, and El over local rings with 1/2

It is proved that every automorphism of an elementary adjoint Chevalley group of type A _l , D _l , or E _l over a local commutative ring with 1/2 is a composition of a ring automorphism and conjugation by some matrix from the normalizer of that Chevalley group in GL(V) (V is an adjoint representation space).     

Doubly Transitive Automorphism Groups of Combinatorial Surfaces

   Abstract. The combinatorial surfaces with doubly transitive automorphism groups are classified. This is established by classifying the automorphism groups of combinatorial surfaces which act doubly transitively on the vertices of the surface. The doubly transitive automorphism groups of combinatorial surfaces are the symmetric group S ₄ , the alternating group A ₅ and the Frobenius group C ₇ · C ₆ . In each case the combinatorial surface is uniquely determined. The symmetric group S ₄ acts doubly transitively on the tetrahedron surface, the alternating group A ₅ on the triangulation of the projective plane with six vertices and the Frobenius group C ₇ · C ₆ on the Moebius torus with seven vertices.     

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.     

Doubly Transitive Automorphism Groups of Combinatorial Surfaces

Abstract. The combinatorial surfaces with doubly transitive automorphism groups are classified. This is established by classifying the automorphism groups of combinatorial surfaces which act doubly transitively on the vertices of the surface. The doubly transitive automorphism groups of combinatorial surfaces are the symmetric group S ₄ , the alternating group A ₅ and the Frobenius group C ₇ · C ₆ . In each case the combinatorial surface is uniquely determined. The symmetric group S ₄ acts doubly transitively on the tetrahedron surface, the alternating group A ₅ on the triangulation of the projective plane with six vertices and the Frobenius group C ₇ · C ₆ on the Moebius torus with seven vertices.     

Bedingungen für die Zweispiegeligkeit der Automorphismengruppen von Cayleyalgebren

In this paper we derive necessary and sufficient conditions for the bireflectionality of the automorphism group of a Cayley algebra over a field of characteristic not 2. These are of particular interest for split Cayley algebras since their automorphism groups are the Chevalley groups of type G ₂. As an application we show the bireflectionality of the automorphism groups of Cayley algebras over real closed fields.     

On the ϕ-structure on the projective groupL 2(q)

In the paper it is proved that the projective groupL ₂(q) cannot be the automorphism group of a finite left distributive quasigroup. This is a special case of the conjecture according to which the automorphism group of a left distributive quasigroup is solvable. Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 725–728, May, 1998.     

Dimension and automorphism groups of lattices

Every group is the automorphism group of a lattice of order dimension at most 4. We conjecture that the automorphism groups of finite modular lattices of bounded dimension do not represent every finite group. It is shown that ifp is a large prime dividing the order of the automorphism group of a finite modular latticeL then eitherL has high order dimension orM _p, the lattice of height 2 and orderp+2, has a cover-preserving embedding inL. We mention a number of open problems. Presented by C. R. Platt.     

Counting Overlattices in Automorphism Groups of Trees

We give an upper bound for the number u _Γ(n) of “overlattices” in the automorphism group of a tree, containing a fixed lattice Γ with index n. For an example of Γ in the automorphism group of a 2p-regular tree whose quotient is a loop, we obtain a lower bound of the asymptotic behavior as well.     

The Classification of Symmetric Transversal Designs <Emphasis Type="Italic">STD</Emphasis><Subscript>4</Subscript>[12; 3]’s

In this article we prove that there is only one symmetric transversal design STD₄[12;3] up to isomorphism. We also show that the order of the full automorphism group of STD₄[12; 3] is 2⁵· 3³ and Aut STD₄[12;3] has a regular subgroup as a permutation group on the point set. We used a computer for our research.Communicated by: C.J. Colbourn     

On automorphisms of finite Abelian <Emphasis Type="Italic">p</Emphasis>-groups

Let A be a finitely generated abelian group. We describe the automorphism group Aut(A) using the rank of A and its torsion part p-part A _p . For a finite abelian p-group A of type (k ₁, ..., k _n ), simple necessary and sufficient conditions for an n × n-matrix over integers to be associated with an automorphism of A are presented. Then, the automorphism group Aut(A) for a finite p-group A of type (k ₁, k ₂) is analyzed. This work has begin during the visit of the second author to the Faculty of Mathematics and Computer Science, Nicolaus Copernicus University during the period July 31–August 13, 2005. This visit was supported by the Nicolaus Copernicus University and a grant from Cnpq.     

Decomposition of Jordan Automorphisms of Strictly Upper Triangular Matrix Algebra Over Commutative Rings

Over a 2-torsionfree commutative ring R with identity, the algebra of all strictly upper triangular n + 1 by n + 1 matrices is denoted by n ₁. In this article, we prove that any Jordan automorphism of n ₁ can be uniquely decomposed as a product of a graph automorphism, a diagonal automorphism, a central automorphism and an inner automorphism for n ≥ 3. In the cases n = 1, 2, we also give a decomposition for any Jordan automorphism of n ₁.     

Real-local fields and the canonical operation of the automorphism group on the space of orderings

The object of our investigation is the canonical operation of the automorphism group of a formally real field F on X_F , the space of orderings of F. For a naturally distinguished class of formally real fields, the so-called real-local fields, the Baer-Krull-bijection induces on X_F the structure of a module over the endomorphism ring of the group of archimedean classes of F. We show that Aut F acts on X_F by affinities with respect to that module structure. Subsequently, this “arithmetization” of the operation is exemplarily applied to the question of transitivity (“When can any two orderings of F be transformed into each other by some automorphism of F?"), and to the investigation of the subgroup of Aut F generated by all order automorphism groups of F.     

Structure of the automorphism group for partially commutative class two nilpotent groups

Let R be a ring, which is either a ring of integers or a field of zero characteristic. For every finite graph Γ, we construct an R-arithmetic linear group H(Γ). The group H(Γ) is realized as the factor automorphism group of a partially commutative class two nilpotent R-group G _Γ. Also we describe the structure of the entire automorphism group of a partially commutative nilpotent R-group of class two.