Gil Kaplan

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.

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.

SOLVABILITY OF FINITE GROUPS VIA CONDITIONS ON PRODUCTS OF 2-ELEMENTS AND ODD p -ELEMENTS

We observe that a solvability criterion for finite groups, conjectured by Miller [The product of two or more groups, Trans. Amer. Math. Soc. 12 (1911)] and Hall [A characteristic property of soluble groups, J. London Math. Soc. 12 (1937)] and proved by Thompson [Nonsolvable finite groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc. 74 (3) (1968)], can be sharpened as follows: a finite group is nonsolvable if and only if it has a nontrivial 2-element and an odd p -element, such that the order of their product is not divisible by either 2 or p . We also prove a solvability criterion involving conjugates of odd p -elements. Finally, we define, via a condition on products of p -elements with p ′ -elements, a formation P p , p ′ , for each prime p . We show that P 2,2 ′ (which contains the odd-order groups) is properly contained in the solvable formation.

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.

Products of Subgroups Which Are Subgroups

We prove conditions for a product of distinct subgroups of an arbitrary group G to be a subgroup of G. In particular, the normal closure of any A ≤ G is equal to the product of some distinct conjugates of A. As an application of the later result we derive constraints on the size of a nontrivial conjugacy class of a finite non-Abelian simple group.

Transversals in permutation groups and factorisations of complete graphs

Let G be a transitive permutation group acting on a finite set of order n. We discuss certain types of transversals for a point stabiliser A in G: free transversals and global transversals. We give sufficient conditions for the existence of such transversals, and show the connection between these transversals and combinatorial problems of decomposing the complete directed graph if* into edge disjoint cycles. In particular, we classify all the inner-transitive Oberwolfach factorisations of the complete directed graph. We mention also a connection to Frobenius theorem.

Nilpotency, Solvability, and the Twisting Function of Finite Groups

Let G be a finite group. For every natural number m, we define the twisting function τ: G m → G m by τ(x 1, ⋅, x m ) = . It is clear that for each normal subset C of G, the function τ: C m → C m is well defined, for each natural number m. Let C be a conjugacy class of size n in G. We prove (Theorem 1) that C is contained in the Fitting subgroup if and only if τ: C m → C m is a permutation, for all 1 ≤ m ≤ 2n. We prove further (Theorem 3) that if τ: G 2 → G 2 is a permutation then G is solvable.

Representation of permutations as products of two cycles

Abstract Given a permutation σ on n letters, we determine for which values of the integers l1 and l2 it is possible to represent σ as a product of two cycles of sizes l1 and l2, respectively. Our results are of a constructive nature. We also deal with the special cases l1=l2 for even permutations and l1=l2+1 for odd permutations, which were solved differently by Bertram in (J. Combin. Theory 12 (1972) 368).

On subgroups containing non-trivial normal subgroups

We prove that ifA≠1 is a subgroup of a finite groupG and the order of an element in the centralizer ofA inG is strictly larger (larger or equal) than the index [G:A], thenA contains a non-trivial characteristic (normal) subgroup ofG. Consequently, ifA is a stabilizer in a transitive permutation group of degreem>1, thenexp(Z(A))<m. These theorems generalize some recent results of Isaacs and the authors.