Abstract. 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.

In this paper we define a new class of partially filled arrays, called relative Heffter arrays, that are a generalization of the Heffter arrays introduced by Archdeacon in 2015. Let v = 2nk + t be a positive integer, where t divides 2nk, and let J be the subgroup of Zv of order t. A Ht(m, n; s, k) Heffter array over Zv relative to J is an m × n partially filled array with elements in Zv such that: (i) each row contains s filled cells and each column contains k filled cells; (ii) for every x ∈ Z 2nk+t \ J, either x or −x appears in the array; (iii) the elements in every row and column sum to 0. In particular, here we study the existence for t = k of integer (i.e. the entries are chosen in ± 1, . . . , 2nk+t 2 and the sums are zero in Z) square relative Heffter arrays.

In the factorial ring of Dirichlet polynomials we explore the connections between how the Dirichlet polynomial P G (s) associated with a finite group G factorizes and the structure of G. If P G (s) is irreducible, then G/Frat G is simple. We investigate whether the converse is true, studying the factorization in the case of some simple groups. For any prime p ≥ 5 we show that if P G (s) = P Alt(p) (s), then G/Frat G ∼ = Alt(p) and P Alt(p) (s) is irreducible. Moreover, if P G (s) = P PSL(2,p) (s), then G/Frat G is simple, but P PSL(2,p) (s) is reducible whenever p = 2 t − 1 and t = 3 mod 4.

We consider finite groups with the property that any proper factor can be generated by a smaller number of elements than the group itself. We study some problems related with the probability of generating these groups with a given number of elements.2000 Mathematics subject classification: primary 20B05, 20P05.

Assume that a finite group G has a unique minimal normal subgroup, say N . We prove that if the order of N is large enough then the following is true: If d randomly chosen elements generate G modulo N , then these elements almost certainly generate G itself.