Arieh Lev

On zero-sum partitions and anti-magic trees

We study zero-sum partitions of subsets in abelian groups, and apply the results to the study of anti-magic trees. Extension to the nonabelian case is also given.

Products of cyclic similarity classes in the groups GLn(F)

Abstract Let G = GL n ( F ) be the group of all n × n invertible matrices over a field F , where ¦ F ¦ ⩾ 4 and n ⩾ 3. Let J 1 , J 2 be two prescribed cyclic elements of G in Jordan canonical form, and assume that all the eigenvalues of J 1 or J 2 lie in F . It is shown that any nonscalar matrix in G can be written as a product of two matrices having the prescribed Jordan canonical forms J 1 , J 2 subject only to the obvious 0 determinant condition. As a corollary we show that if ¦ F ¦ ⩾ 4, the group PSL n ( F ) has a conjugacy class C such that C 2 = PSL n ( F ). Another corollary states that every nonscalar matrix in G can be written as a product of two cyclic unipotent matrices.

Regular Oberwolfach Problems and Group Sequencings

We deal with Oberwolfach factorizations of the complete graphs Kn and K*n, which admit a regular group of automorphisms. We show that the existence of such a factorization is equivalent to the existence of a certain difference sequence defined on the elements of the automorphism group, or to a certain sequencing of the elements of that group. In the particular case of a hamiltonian factorization of the directed graph K*n which admits a regular group of automorphisms G (|G|=n?1), we have that such a factorization exists if and only if G is sequenceable. We shall demonstrate how the mentioned above (difference) sequences may be used in the construction of such factorizations. We prove also that a hamiltonian factorization of the undirected graph Kn (n odd) which admits a regular group of automorphisms G (|G|=(n?1)/2) exists if and only if n?3 (mod4), without further restrictions on the structure of G.

Products of cyclic conjugacy classes in the groups PSL(n,F)

Abstract Let G be the group PSL(n,F), where F is a field and n ⩾ 3. If C is a conjugacy class of G, denote C-1 = {c-1∥c ∈ C}. The following is proved: (1) If C1, C2, C3 are cyclic conjugacy classes of G, then C1C2C3 ⊇ G - {1G}. (2) If F is algebraically closed and C1, C2 are cyclic conjugacy classes of G, then C1C2 = G if and only if C1 = C-12. Generalizations of these results, concerning factorizations of a given nonscalar invertible matrix as a product of two or three cyclic matrices, each lying in a prescribed conjugacy class of GL(n, F) [or SL(n,F], are discussed.

Bases and decomposition numbers of finite groups

G. The minimal cardinality of a basis of G is denoted by r(G). A family of finite groups ,3 is well-based if there exists a constant c such that r(G) < c [GI 1/z for each G ~ .3. The problem of estimating r(G) for cyclic groups was first proposed by I. Schur and various bounds were obtained by Rohrbach [7], Moser [5], St6hr [9], Klotz [3] and others. Bases for arbitrary groups were dealt by Rohrbach [8] and lately by Bertram and Herzog [1] and Nathanson [6]. In [8] Rohrbach showed that the class of abelian groups with a bounded number of generators is well-based. He also mentioned that the class of solvable groups which possess a series of a bounded length with cyclic factors is well- based. In [1] Bertram and Herzog showed that the families of the nilpotent groups, as well as the families of the alternating and symmetric groups, are well-based. In [6] Nathanson showed that r(G) < 2(IGI loglG[) 1/2 + 2 for every finite group G of order n. In this paper we prove that the family of all finite groups is well-based, with r (G) < ~IGI

Powers of 1-cyclic conjugacy classes in the groups GLn(F) and SLn(F)

Abstract Let A ∈ GL n ( F ), where F is a field. We say that A is 1- cyclic if A is similar to a matrix of the form A ′ = diag{ A 1 , A 2 , …, A k }, where A i ∈ GL l i ( F ) is cyclic for 1 ⩽ i ⩽ k , l 1 ∈ {0, 1}, and l i ⩾ 2 for 2 ⩽ i ⩽ k . It is shown that if A ∈ GL n ( F ) is 1-cyclic, where n ⩾ 2 and | F | ⩾ 4, then every nonscalar matrix M ∈ GL n ( F ) whose determinant equals (det A ) 4 is the product of four matrices which are similar to A under matrices of SL n ( F ). The problem of expressing a scalar matrix as a product of similar 1-cyclic matrices is also discussed. The above result is applied to problems of factorizing matrices in the group SL n ( F ) into products of unipotent matrices of index 2, and into products of matrices of (fixed) finite order.

Upper bounds on the automorphism group of a graph

We give upper bounds on the order of the automorphism group of a simple graph In this note we present some upper bounds on the order of the automorphism group of a graph, which is assumed to be simple, having no loops or multiple edges. Somewhat surprisingly, we did not find such bounds in the literature and the goal of this paper is to fill this gap. As a matter of fact, implicitly such bounds were contained in works dealing with the edge reconstruction conjecture and are the corollaries of a simple theorem which is presented below (Theorem 1). Therefore we bring together a few results spread in different, sometimes in difficult to reach, sources (see Theorem 2 below). In Theorem 3

On large subgroups of finite groups

The proof of the theorem uses the classification of the finite simple groups. Note that the result of the theorem is best possible since all proper non-trivial subgroups of a group of order p2, p a prime, are of order p. The notation is standard. For a group G, JG/ is the order of G, and K< G means K is a subgroup of G. For the finite groups of Lie type, the notation of [Cl] is used. The author is grateful to Gady Kozma and to the referee for their constructive remarks.

Bertrand's postulate, the prime number theorem and product anti-magic graphs

Let the edges of a finite simple graph G=(V,E),|V|=n,|E|=m, be labeled by 1,2,...,m. Denote by w(u) the product of all the labels of edges incident with a vertex u. The graph G is called product anti-magic if it is possible that the above labeling results in all values w(u) being distinct. Following an old conjecture of Hartsfield and Ringel on (sum) anti-magic graphs (see [N. Hartsfield and G. Ringel, Pearls in Graph Theory, Academic Press, Inc., Boston, 1990, pp. 108-109 (revised version, 1994)]), Figueroa-Centeno et al. [Bertrands postulate and magical product labellings, Bull. ICA 30 (2000) 53-65] conjectured that every connected graph of size m is product anti-magic iffm>=3. In this paper we prove this conjecture for dense graphs, complete multi-partite graphs and some other families of graphs.

On the Dimension and Basis Concepts in Finite Groups

Abstract We study the generation of a finite group by its conjugacy classes, while generalizing basic concepts from linear algebra: basis and dimension. Besides the well known Burnside Basis Theorem for finite p-groups, there is no direct extension of these concepts to other families of finite groups. We show that by considering generating sets consisting of conjugacy classes, there is a possibility for such a generalization.