Let G be a finite cyclic group. Every sequence S over G can be written in the form $S=\left(n₁g\right)·...·\left(n_{l}g\right)$ where g ∈ G and $n₁,...,n_{l}i\in [1,ord\left(g\right)]$, and the index ind(S) is defined to be the minimum of $(n₁+\cdots +n_l)/ord\left(g\right)$ over all possible g ∈ G such that ⟨g⟩ = G. A conjecture says that every minimal zero-sum sequence of length 4 over a finite cyclic group G with gcd(|G|,6) = 1 has index 1. This conjecture was confirmed recently for the case when |G| is a product of at most two prime powers. However, the general case is still open. In this paper, we make some...

Let$$T\left(x\right)=K_{1}x{log}^{2}x+K_{2}xlogx+K_{3}x+\Delta \left(x\right),$$where $T\left(x\right)$ denotes the number of subgroups of all abelian groups whose order does not exceed $x$ and whose rank does not exceed $2$, and $\Delta \left(x\right)$ is the error term. It is proved that$${\int }_1^X{\Delta }^2\left(x\right)dx\ll X^2{log}^{31/3}X,{\int }_1^X{\Delta }^2\left(x\right)dx=\Omega \left(X^2{log}^{4}X\right).$$

A subset $S$ of a finite abelian group, written additively, is called zero-sumfree if the sum of the elements of each non-empty subset of $S$ is non-zero. We investigate the maximal cardinality of zero-sumfree sets, i.e., the (small) Olson constant. We determine the maximal cardinality of such sets for several new types of groups; in particular, $p$-groups with large rank relative to the exponent, including all groups with exponent at most five. These results are derived as consequences of more general...

The investigation of quantitative aspects of non-unique factorizations in the ring of integers of an algebraic number field gives rise to combinatorial problems in the class group of this number field. In this paper we investigate the combinatorial problems related to the function 𝓟(H,𝓓,M)(x), counting elements whose sets of lengths have period 𝓓, for extreme choices of 𝓓. If the class group meets certain conditions, we obtain the value of an exponent in the asymptotic formula of this function...

The paper extends the results given by M. Křížek and L. Somer, On a connection of number theory with graph theory, Czech. Math. J. 54 (129) (2004), 465–485 (see [5]). For each positive integer $n$ define a digraph $\Gamma \left(n\right)$ whose set of vertices is the set $H=\{0,1,\cdots ,n-1\}$ and for which there is a directed edge from $a\in H$ to $b\in H$ if $a^3\equiv b(modn).$ The properties of such digraphs are considered. The necessary and the sufficient condition for the symmetry of a digraph $\Gamma \left(n\right)$ is proved. The formula for the number of fixed points of $\Gamma \left(n\right)$ is established....

Let $G\cong C_{n_1}\oplus ...\oplus C_{n_r}$ be a finite and nontrivial abelian group with $n_1|n_2|...|n_r$. A conjecture of Hamidoune says that if $W=w_1·...·w_n$ is a sequence of integers, all but at most one relatively prime to $\left|G\right|$, and $S$ is a sequence over $G$ with $\left|S\right|\ge \left|W\right|+\left|G\right|-1\ge \left|G\right|+1$, the maximum multiplicity of $S$ at most $\left|W\right|$, and $\sigma \left(W\right)\equiv 0mod\left|G\right|$, then there exists a nontrivial subgroup $H$ such that every element $g\in H$ can be represented as a weighted subsequence sum of the form $g=\underset{i=1}{\sum ^n}{w}_i{s}_i$, with $s_1·...·s_n$ a subsequence of $S$. We give two examples showing this does not hold in general, and characterize the counterexamples...

We present a system of interrelated conjectures which can be considered as restricted addition counterparts of classical theorems due to Kneser, Kemperman, and Scherk. Connections with the theorem of Cauchy-Davenport, conjecture of Erdős-Heilbronn, and polynomial method of Alon-Nathanson-Ruzsa are discussed.The paper assumes no expertise from the reader and can serve as an introduction to the subject.