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...

It is a well-known fact that modules over a commutative ring in general cannot be classified, and it is also well-known that we have to impose severe restrictions on either the ring or on the class of modules to solve this problem. One of the restrictions on the modules comes from freeness assumptions which have been intensively studied in recent decades. Two interesting, distinct but typical examples are the papers by Blass [1] and Eklof [8], both jointly with Shelah. In the first case the authors...

This paper investigates the productivity of the Zariski topology ${ℨ}_G$ of a group $G$. If $𝒢=\{G_i\mid i\in I\}$ is a family of groups, and $G={\prod }_{i\in I}{G}_i$ is their direct product, we prove that ${ℨ}_G\subseteq {\prod }_{i\in I}{ℨ}_{G_i}$. This inclusion can be proper in general, and we describe the doubletons $𝒢=\{G_1,G_2\}$ of abelian groups, for which the converse inclusion holds as well, i.e., ${ℨ}_G={ℨ}_{G_1}\times {ℨ}_{G_2}$. If $e_2\in G_2$ is the identity element of a group $G_2$, we also describe the class $\Delta$ of groups $G_2$ such that $G_1\times \left\{e_2\right\}$ is an elementary algebraic subset of $G_1\times G_2$ for every group $G_1$. We show among others, that $\Delta$ is stable...

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....

In this paper, we give a generalization of Baer Theorem on the injective property of divisible abelian groups. As consequences of the obtained result we find a sufficient condition for a group $G$ to express as semi-direct product of a divisible subgroup $D$ and some subgroup $H$. We also apply the main Theorem to the $p$-groups with center of index $p^2$, for some prime $p$. For these groups we compute $N_c\left(G\right)$ the number of conjugacy classes and $N_a$ the number of abelian maximal subgroups and $N_{na}$ the number of nonabelian...

Let $\lambda$ be an infinite cardinal. Set ${\lambda }_0=\lambda$, define ${\lambda }_{i+1}=2^{{\lambda }_i}$ for every $i=0,1,\cdots$, take $\mu$ as the first cardinal with ${\lambda }_i<\mu$, $i=0,1,\cdots$ and put $\kappa ={\left({\mu }^{{\aleph }_0}\right)}^{+}$. If $F$ is a torsion-free group of cardinality at least $\kappa$ and $K$ is its subgroup such that $F/K$ is torsion and $|F/K|\le \lambda$, then $K$ contains a non-zero subgroup pure in $F$. This generalizes the result from a previous paper dealing with $F/K$$p$-primary.