Irreducible p-constant characters of finite reflection groups

A complex irreducible character of a finite group G is said to be p-constant, for some prime p dividing the order of G, if it takes constant value at the set of p-singular elements of G. In this paper we classify irreducible p-constant characters for finite reflection groups, nilpotent groups and complete monomial groups. We also propose some conjectures about the structure of the groups admitting such characters.Comment: To appear in Journal of Group Theor

The (2,3)-generation of the special linear groups over finite fields

We complete the classification of the finite special linear groups \SL_n(q) which are $(2,3)$-generated, i.e., which are generated by an involution and an element of order $3$. This also gives the classification of the finite simple groups \PSL_n(q) which are $(2,3)$-generated.Comment: 5 page

The (2,3)-generation of the classical simple groups of dimension 6 and 7

In this paper we prove that the finite simple groups $PSp_6(q)$, $\Omega_7(q)$ and $PSU_7(q^2)$ are (2,3)-generated for all q. In particular, this result completes the classification of the (2,3)-generated finite classical simple groups up to dimension 7

A generalization of the problem of Mariusz Meszka

Mariusz Meszka has conjectured that given a prime p=2n+1 and a list L containing n positive integers not exceeding n there exists a near 1-factor in K_p whose list of edge-lengths is L. In this paper we propose a generalization of this problem to the case in which p is an odd integer not necessarily prime. In particular, we give a necessary condition for the existence of such a near 1-factor for any odd integer p. We show that this condition is also sufficient for any list L whose underlying set S has size 1, 2, or n. Then we prove that the conjecture is true if S={1,2,t} for any positive integer t not coprime with the order p of the complete graph. Also, we give partial results when t and p are coprime. Finally, we present a complete solution for t<12.Comment: 15 page

A problem on partial sums in abelian groups

In this paper we propose a conjecture concerning partial sums of an arbitrary finite subset of an abelian group, that naturally arises investigating simple Heffter systems. Then, we show its connection with related open problems and we present some results about the validity of these conjectures

Globally simple Heffter arrays and orthogonal cyclic cycle decompositions

In this paper we introduce a particular class of Heffter arrays, called globally simple Heffter arrays, whose existence gives at once orthogonal cyclic cycle decompositions of the complete graph and of the cocktail party graph. In particular we provide explicit constructions of such decompositions for cycles of length $k\leq 10$. Furthermore, starting from our Heffter arrays we also obtain biembeddings of two $k$-cycle decompositions on orientable surfaces.Comment: The present version also considers the problem of biembedding

Emergent Noncommutative gravity from a consistent deformation of gauge theory

Starting from a standard noncommutative gauge theory and using the Seiberg-Witten map we propose a new version of a noncommutative gravity. We use consistent deformation theory starting from a free gauge action and gauging a killing symmetry of the background metric to construct a deformation of the gauge theory that we can relate with gravity. The result of this consistent deformation of the gauge theory is nonpolynomial in A_\mu. From here we can construct a version of noncommutative gravity that is simpler than previous attempts. Our proposal is consistent and is not plagued with the problems of other approaches like twist symmetries or gauging other groups.Comment: 18 pages, references added, typos fixed, some concepts clarified. Paragraph added below Eq. (77). Match published PRD version

Some new results about a conjecture by Brian Alspach

In this paper we consider the following conjecture, proposed by Brian Alspach, concerning partial sums in finite cyclic groups: given a subset $A$ of $\mathbb{Z}_n\setminus \{0\}$ of size $k$ such that $\sum_{z\in A} z\not= 0$, it is possible to find an ordering $(a_1,\ldots,a_k)$ of the elements of $A$ such that the partial sums $s_i=\sum_{j=1}^i a_j$, $i=1,\ldots,k$, are nonzero and pairwise distinct. This conjecture is known to be true for subsets of size $k\leq 11$ in cyclic groups of prime order. Here, we extend such result to any torsion-free abelian group and, as a consequence, we provide an asymptotic result in $\mathbb{Z}_n$. We also consider a related conjecture, originally proposed by Ronald Graham: given a subset $A$ of $\mathbb{Z}_p\setminus\{0\}$, where $p$ is a prime, there exists an ordering of the elements of $A$ such that the partial sums are all distinct. Working with the methods developed by Hicks, Ollis and Schmitt, based on the Alon's combinatorial Nullstellensatz, we prove the validity of such conjecture for subsets $A$ of size $12$