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.

An exact sequence $0\to A\to B\to C\to 0$ of torsion-free abelian groups is quasi-balanced if the induced sequence $$0\to 𝐐\otimes Hom(X,A)\to 𝐐\otimes Hom(X,B)\to 𝐐\otimes Hom(X,C)\to 0$$ is exact for all rank-1 torsion-free abelian groups $X$. This paper sets forth the basic theory of quasi-balanced sequences, with particular attention given to the case in which $C$ is a Butler group. The special case where $B$ is almost completely decomposable gives rise to a descending chain of classes of Butler groups. This chain is a generalization of the chain of Kravchenko classes that arise from balanced...