The investigation of certain counting functions of elements with given factorization properties in the ring of integers of an algebraic number field gives rise to combinatorial problems in the class group. In this paper a constant arising from the investigation of the number of algebraic integers with factorizations of at most k different lengths is investigated. It is shown that this constant is positive if k is greater than 1 and that it is also positive if k equals 1 and the class group satisfies...

The investigation of the counting function of the set of integral elements, in an algebraic number field, with factorizations of at most k different lengths gives rise to a combinatorial constant depending only on the class group of the number field and the integer k. In this paper the value of these constants, in case the class group is an elementary p-group, is estimated, and determined under additional conditions. In particular, it is proved that for elementary 2-groups these constants are equivalent...

For a finite abelian group G and a splitting field K of G, let (G,K) denote the largest integer l ∈ ℕ for which there is a sequence $S=g₁·...·g_l$ over G such that $(X^{g₁}-a₁)·...·(X^{g_l}-a_l)\ne 0\in K\left[G\right]$ for all $a₁,...,a_l\in K^{\times }$. If (G) denotes the Davenport constant of G, then there is the straightforward inequality (G) - 1 ≤ (G,K). Equality holds for a variety of groups, and a conjecture of W. Gao et al. states that equality holds for all groups. We offer further groups for which equality holds, but we also give the first examples of groups G for which (G) -...

A power digraph, denoted by $G(n,k)$, is a directed graph with ${\mathbb{Z}}_n=\{0,1,\cdots ,n-1\}$ as the set of vertices and $E=\{(a,b):a^k\equiv b(modn)\}$ as the edge set. In this paper we extend the work done by Lawrence Somer and Michal Křížek: On a connection of number theory with graph theory, Czech. Math. J. 54 (2004), 465–485, and Lawrence Somer and Michal Křížek: Structure of digraphs associated with quadratic congruences with composite moduli, Discrete Math. 306 (2006), 2174–2185. The heights of the vertices and the components of $G(n,k)$ for $n\ge 1$ and $k\ge 2$ are determined....

Let X be a countable discrete Abelian group, Aut(X) the set of automorphisms of X, and I(X) the set of idempotent distributions on X. Assume that α₁, α₂, β₁, β₂ ∈ Aut(X) satisfy $\beta ₁\alpha {₁}^{-1}\pm \beta ₂\alpha {₂}^{-1}\in Aut\left(X\right)$. Let ξ₁, ξ₂ be independent random variables with values in X and distributions μ₁, μ₂. We prove that the symmetry of the conditional distribution of L₂ = β₁ξ₁ + β₂ξ₂ given L₁ = α₁ξ₁ + α₂ξ₂ implies that μ₁, μ₂ ∈ I(X) if and only if the group X contains no elements of order two. This theorem can be considered as an analogue...

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...

In this paper, the structures of collection of pronormal subgroups of dicyclic, symmetric and alternating groups $G$ are studied in respect of formation of lattices $\mathrm{L}\left(G\right)$ and sublattices of $\mathrm{L}\left(G\right)$. It is proved that the collections of all pronormal subgroups of ${\mathrm{A}}_n$ and S${}_n$ do not form sublattices of respective $\mathrm{L}\left({\mathrm{A}}_n\right)$ and $\mathrm{L}\left({\mathrm{S}}_n\right)$, whereas the collection of all pronormal subgroups $\mathrm{LPrN}\left({\mathrm{Dic}}_n\right)$ of a dicyclic group is a sublattice of $\mathrm{L}\left({\mathrm{Dic}}_n\right)$. Furthermore, it is shown that $\mathrm{L}\left({\mathrm{Dic}}_n\right)$ and $\mathrm{LPrN}({\mathrm{Dic}}_n$) are lower semimodular lattices.

Motivated by the paper by H. Herrlich, E. Tachtsis (2017) we investigate in ZFC the following compactness question: for which uncountable cardinals $\kappa$, an arbitrary nonempty system $S$ of homogeneous $\mathbb{Z}$-linear equations is nontrivially solvable in $\mathbb{Z}$ provided that each of its subsystems of cardinality less than $\kappa$ is nontrivially solvable in $\mathbb{Z}$?