Narrow Results

In a paper concerning the relationship between coding theory and factorization theory Restivo, Salemi, and Sportelli made a conjecture that if subsets possess certain properties then they cannot form a factorization of a finite cyclic group. In this note it is shown that in fact such factorizations do exist.

We investigate the descriptive complexity of finite abelian groups. Using Ehrenfeucht–Fraïssé games we find upper and lower bounds on quantifier depth, quantifier alternations, and number of variables of a first-order sentence that distinguishes two finite abelian groups. Our main results are the following. Let G₁ and G₂ be a pair of non-isomorphic finite abelian groups. Then there exists a positive integer m that divides one of the two groups' orders such that the following holds: (1) there exists a first-order sentence φ that distinguishes G₁ and G₂ such that φ is existential, has quantifier depth O(log m), and has at most 5 variables and (2) if φ is a sentence that distinguishes G₁ and G₂ then φ must have quantifier depth Ω(log m). These results are applied to (1) get bounds on the first-order distinguishability of dihedral groups, (2) to prove that on the class of finite groups both cyclicity and the closure of a single element are not first-order definable, and (3) give a different proof for the first-order undefinability of simplicity, nilpotency, and the normal closure of a single element on the class of finite groups (their undefinability were shown by Koponen and Luosto in an unpublished paper).

Let $A$ be a nilpotent $p$-group of finite exponent and $B$ be an abelian $p$-group of finite exponent for a given prime number $p$. Then the wreath product $A\mathrm{\text{Wr}}B$ generates the variety $\mathrm{\text{var}}(A)\mathrm{\text{var}}(B)$ if and only if the group $B$ contains a subgroup isomorphic to the direct product $C_{p^v}^{\infty }$ of countably many copies of the cycle $C_{p^v}$ of order $p^v=\mathrm{\text{exp}}(B)$. The obtained theorem continues our previous study of cases when $\mathrm{\text{var}}(A\mathrm{\text{Wr}}B)=\mathrm{\text{var}}(A)\mathrm{\text{var}}(B)$ holds for some other classes of groups $A$ and $B$ (abelian groups, finite groups, etc.).

We prove that abelian subgroups of the outer automorphism group of a free group are quasi-isometrically embedded. Our proof uses recent developments in the theory of train track maps by Feighn–Handel. As an application, we prove the rank conjecture for $\text{Out}(F_n)$.

In connection to two recent publications of ours in Arch. Math. Basel (2021) and Acta Math. Hungar. (2022), respectively, and in regard to the results obtained in Arch. Math. Basel (2012), we have the motivation to study the near property of both Bassian and generalized Bassian groups. Concretely, we prove that if an arbitrary reduced group has the property that all of its proper subgroups are generalized Bassian, then it is also generalized Bassian itself, that is, if G is an arbitrary nearly generalized Bassian group, then either G is a quasi-cyclic group or G is generalized Bassian of finite torsion-free rank. Moreover, we show that there is a complete description of the so-called super Bassian groups that are those groups whose epimorphic images are all Basian, and give their full characterization. Also, we establish that the hereditary property of Bassian groups gives nothing new as it coincides with the ordinary Bassian property.

We characterize rings over which every cotorsion module is pure injective (Xu rings) in terms of certain descending chain conditions and the Ziegler spectrum, which renders the classes of von Neumann regular rings and of pure semisimple rings as two possible extremes. As preparation, descriptions of pure projective and Mittag–Leffler preenvelopes with respect to so-called definable subcategories and of pure generation for such are derived, which may be of interest on their own. Infinitary axiomatizations lead to coherence results previously known for the special case of flat modules. Along with pseudoflat modules we introduce quasiflat modules, which arise naturally in the model-theoretic and the category-theoretic contexts.

An n-ary endofunction on an abelian group G is a function f : Gⁿ → G such that f(θg₁,…,θg_n) = θ f(g₁,…,g_n) for all endomorphisms θ of G. A group G is endoprimal if, for each natural number n, each n-ary endofunction has the following simple form: for some collection of integers {l_i : 1 ≤ i ≤ n}. The notion of endoprimality arises from universal algebra in a natural way and has been applied to the study of abelian groups in papers Davey and Pitkethly (97), Kaarli and Marki (99) and Göbel, Kaarli, Marki, and Wallutis (to appear). These papers make the case that the notion of endoprimality can give rise to interesting and tractable classes of abelian groups. We continue working along these lines, adapting our definition to make it more suitable for working with general classes of abelian groups. We study generalized endoprimal (ge) abelian groups. Here every n-ary endofunction is required to be of the form for some collection of central endomorphisms {λ_i : 1 ≤ i ≤ n} of G. (Note that such a sum is always an endofunction.) We characterize generalized endoprimal abelian groups in a number of cases, in particular for torsion groups, torsion-free finite rank groups G such that E(G) has zero nil radical, and self-small mixed groups of finite torsion-free rank.

Let F be a field and G an Abelian group. For every prime number q and every ordinal number α we compute only in terms of F and G the Warfield q-invariants W_{α, q}(VF[G]) of the group VF[G] of all normed units in the group algebra F[G] under some minimal restrictions on F and G.

This expands own recent results from (Extracta Mathematicae, 2005) and (Collectanea Mathematicae, 2008).

Let R be a commutative unital ring of arbitrary characteristic and let G be a multiplicative Abelian group. For the group ring RG we completely calculate the number (finite or infinite) of its idempotents only in terms of R, G and their sections. This strengthens our previous results in Sarajevo J. Math. (2011) and Filomat (2012).

Let ${𝒮}$ denote the set of positive integers that may appear as the real genus of a finite abelian group. We obtain a set of (simple) necessary conditions for an integer $g$ to belong to ${𝒮}$. We also prove that the real genus of an abelian group is not congruent to 3 modulo 4 and that the genus of an abelian group of odd order is a multiple of 4. Finally, we obtain upper and lower bounds for the density of the set ${𝒮}$.

In this paper, we address the question of finding the length of group algebras of finite abelian groups in the case when the characteristic of the ground field divides the order of the group. We evaluate the exact length for an arbitrary abelian $p$-group. For other groups we provide upper and lower bounds for the length of their group algebras.

In this paper we solve the problem of finding the length of group algebras of arbitrary finite abelian groups in the case when the characteristic of the ground field does not divide the order of the group. We show that these group algebras have maximal possible lengths for infinite fields and sufficiently large finite fields since they are one-generated. In case of small fields we prove that the length is bounded from above by a logarithmic function of the order of the group.

The subgroup $G\le A$ is absolute direct summand (ADS) if, for every $G$-high subgroup $H\le A$ (i.e. maximal with respect to the property $G\cap H=0$), we have $A=G\oplus H$, and $A$ itself is an ADS group if all of its summands inherit this property. In this paper, we characterize the structure of ADS abelian groups, which can be defined as groups such that members of the decomposition into two direct summands are mutually injective. In particular, an ADS abelian group is either divisible, or a direct sum of an indecomposable torsion-free group and a divisible torsion group, or a torsion group such that each $p$-component is a direct sum of cyclic $p$-groups of the same length or of Prüfer groups.

We define and study the so-called CI-extending property for Abelian groups, thus somewhat extending results due to Birkenmeier et al. in Commun. Algebra (2001) concerning the FI-extending property.

A finite group is said to be weakly separable if every algebraic isomorphism between two $S$-ringsover this group is induced by a combinatorial isomorphism. We prove that every abelian weakly separable group only belongs to one of several explicitly given families.

Let $G$ be a finite abelian group and $S$ be a subset of $G$. Let $\mathrm{\Sigma }(S)$ denote the set of group elements which can be expressed as a sum of a nonempty subset of $S$. We say that $S$ is zero-sum free if $0\notin \mathrm{\Sigma }(S)$. In this paper, we show that if $S$ is zero-sum free with $\left|S\right|\ge 6$, then $|\mathrm{\Sigma }(S)|\ge 3\left|S\right|$. Furthermore, if $S$ is zero-sum free and $\left|S\right|\ge 7$, with the assumption that the subgroup generated by $S$ is noncyclic and the order $|〈S〉|\ge 146$, we are able to show that $|\mathrm{\Sigma }(S)|\ge 4\left|S\right|-3$.

A problem for Abelian groups is formulated with motivations from the theory of constant weight codes. The problem is solved for the case (ℤ₂)^r.