Related papers: On the structure of zero-sum free set with minimum…

For which groups $G$ is it true that for all fields $k$, every non-monomial element of the group algebra $k\,G$ generates a proper $2$-sided ideal? The only groups for which we know this are the torsion-free abelian groups. We would like to…

This survey article has two components. The first part gives a gentle introduction to Serre's notion of $G$-complete reducibility, where $G$ is a connected reductive algebraic group defined over an algebraically closed field. The second…

We prove that a random group, in Gromov's density model with $d<1/16$, satisfies a universal sentence $\sigma$ (in the language of groups) if and only if $\sigma$ is true in a nonabelian free group.

Let $G$ be a finite abelian group with $\exp(G)$ the exponent of $G$. Then $\mathsf W(G)$ denotes the set of cross numbers of minimal zero-sum sequences over $G$ and $\mathsf w(G)$ denotes the set of all cross numbers of non-trivial…

For an arbitrary group $G$, it is shown that either the semigroup rank $G{\rm rk}S$ equals the group rank $G{\rm rk}G$, or $G{\rm rk}S = G{\rm rk}G+1$. This is the starting point for the rest of the article, where the semigroup rank for…

Let $k$ be a field of characteristic different from $2$ and let $G$ be a nonabelian residually torsion-free nilpotent group. It is known that $G$ is an orderable group. Let $k(G)$ denote the subdivision ring of the Malcev-Neumann series…

Given a saturated fusion system $\mathcal{F}$ over a $2$-group $S$, we prove that $S$ is abelian provided any element of $S$ is $\mathcal{F}$-conjugate to an element of $Z(S)$. This generalizes a Theorem of Camina--Herzog, leading to a…

Given a finite subset A of an abelian group G, we study the set k \wedge A of all sums of k distinct elements of A. In this paper, we prove that |k \wedge A| >= |A| for all k in {2,...,|A|-2}, unless k is in {2,|A|-2} and A is a coset of an…

A numerical set $S$ with Frobenius number $g$ is a set of integers with $\min(S) = 0$ and $\max(\Zbb - S)=g$, and its atom monoid is $A(S) = \setpres{n \in \Zbb}{$n+s \in S$ for all $s \in S$}$. Let $\gamma_g$ be the number of numerical…

Let $G$ be an abelian group, and let $\mathcal F (G)$ be the free commutative monoid with basis $G$. For $\Omega \subset \mathcal F (G)$, define the universal zero-sum invariant ${\mathsf d}_{\Omega}(G)$ to be the smallest integer $\ell$…

Let $\Gamma$ be a $T$-ideal of identities of an affine PI-algebra over an algebraically closed field $F$ of characteristic zero. Consider the family $\mathcal{M}_{\Gamma}$ of finite dimensional algebras $\Sigma$ with $Id(\Sigma) = \Gamma$.…

Let $\mathcal{S}$ be a finite semigroup, and let $E(\mathcal{S})$ be the set of all idempotents of $\mathcal{S}$. Gillam, Hall and Williams proved in 1972 that every $\mathcal{S}$-valued sequence $T$ of length at least…

Let $G$ be a finite group. We let $\f{m}(G)$ and $\sig(G)$ denote the number of maximal subgroups of $G$ and the least positive integer $n$ such that $G$ is written as the union of $n$ proper subgroups, respectively. In this paper we…

Let G be an abelian group. For a subset A of G, Cyc(A) denotes the set of all elements x of G such that the cyclic subgroup generated by x is contained in A, and G is said to have the small subgroup generating property (abbreviated to SSGP)…

Zero-sum problems for abelian groups and covers of the integers by residue classes, are two different active topics initiated by P. Erdos more than 40 years ago and investigated by many researchers separately since then. In an earlier…

Let $G$ be a finite group and $\sigma_1(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$. In this note, we prove that if $\sigma_1(G)<\frac{117}{20}$, then $G$ is solvable. Moreover, we have $\sigma_1(G)=\frac{117}{20}$ if and only if $G\cong A_5$.…

Peterzil and Steinhorn proved that if a group $G$ definable in an $o$-minimal structure is not definably compact, then $G$ contains a definable torsion-free subgroup of dimension one. We prove here a $p$-adic analogue of the…

For a finite abelian group $G$ and a positive integer $k$, let $s_{k}(G)$ denote the smallest integer $\ell\in\mathbb{N}$ such that any sequence $S$ of elements of $G$ of length $|S|\geq\ell$ has a zero-sum subsequence with length $k$. The…

Let $p > 155$ be a prime and let $G$ be a cyclic group of order $p$. Let $S$ be a minimal zero-sum sequence with elements over $G$, i.e., the sum of elements in $S$ is zero, but no proper nontrivial subsequence of $S$ has sum zero. We call…

Let G be any abelian group and {a_sG_s}_{s=1}^k be a finite system of cosets of subgroups G_1,...,G_k. We show that if {a_sG_s}_{s=1}^k covers all the elements of G at least m times with the coset a_tG_t irredundant then [G:G_t]\le 2^{k-m}…