Zero-sum problems in finite abelian groups: A survey

We give an overview of zero-sum theory in finite abelian groups, a subfield of additive group theory and combinatorial number theory. In doing so we concentrate on the algebraic part of the theory and on the development since the appearance of the survey article by Y. Caro in 1996.     

On the order of elements in long minimal zero-sum sequences

Let G be a finite abelian group and S = a minimal zero-sum sequence in G of maximal length |S| = l. We study the order of the elements      

Normal sequences over finite abelian groups

In this note, we obtain the structure of short normal sequences over a finite abelian p-group or a finite abelian group of rank two, thus answering positively a conjecture of Gao and Zhuang for various groups. The results obtained here improve all known results on this conjecture.     

On zero-sum subsequences of restricted size. IV

Summary For a finite abelian group G, we investigate the invariant s(G) (resp. the invariant s₀(G)) which is defined as the smallest integer l N such that every sequence S in G of length |S| l has a subsequence T with sum zero and length |T|= exp(G) (resp. length |T|0 mod exp(G)).     

Distinct length modular zero-sum subsequences: A proof of Graham's conjecture

Text

Let S be a sequence of n nonnegative integers not exceeding n−1 such that S takes at least three distinct values. We show that S has two nonempty zero-sum subsequences with distinct lengths. This proves a conjecture of R.L. Graham. The validity of this conjecture was verified by Erd?s and Szemerédi for all sufficiently large prime n.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=LftJj-E6aQA.     

Scenery reconstruction on finite abelian groups

We consider the question of when a random walk on a finite abelian group with a given step distribution can be used to reconstruct a binary labeling of the elements of the group, up to a shift. Matzinger and Lember (2006) give a sufficient condition for reconstructability on cycles. While, as we show, this condition is not in general necessary, our main result is that it is necessary when the length of the cycle is prime and larger than 5, and the step distribution has only rational probabilities. We extend this result to other abelian groups.     

On the index of minimal zero-sum sequences over finite cyclic groups

Let G be a cyclic group of order n?2 and a sequence over G. We say that S is a zero-sum sequence if and that S is a minimal zero-sum sequence if S is a zero-sum sequence and S contains no proper zero-sum sequence.The notion of the index of a minimal zero-sum sequence (see Definition 1.1) in G has been recently addressed in the mathematical literature. Let l(G) be the smallest integer t∈N such that every minimal zero-sum sequence S over G with length |S|?t satisfies index(S)=1. In this paper, we first prove that for n?8. Secondly, we obtain a new result about the multiplicity and the order of elements in long zero-sumfree sequences.     

Counting subset sums of finite abelian groups

In this paper, we obtain an explicit formula for the number of zero-sum k-element subsets in any finite abelian group.     

Quadratic characters in groups of odd order

We prove that in a finite group of odd order, the number of irreducible quadratic characters is the number of quadratic conjugacy classes.     

Direct sums of local torsion-free abelian groups

The category of local torsion-free abelian groups of finite rank is known to have the cancellation and -th root properties but not the Krull-Schmidt property. It is shown that 10 is the least rank of a local torsion-free abelian group with two non-equivalent direct sum decompositions into indecomposable summands. This answers a question posed by M.C.R. Butler in the 1960's.

     

Zero-sum invariants on finite abelian groups with large exponent

Let $G$ be an additive finite abelian group with exponent $n$. Let $\mathsf{D}\left(G\right)$ be the Davenport constant of $G$, ${\mathsf{s}}_{kn}\left(G\right)$ the $k$th Erd?s–Ginzburg–Ziv constant of $G$, where $k$ is a positive integer. Recently, Gao, Han, Peng and Sun conjectured that ${\mathsf{s}}_{kn}\left(G\right)=kn+\mathsf{D}\left(G\right)?1$ holds if $k\ge ?\frac{\mathsf{D}\left(G\right)}{n}?$. Let $m,n$ be positive integers and $H$ an abelian $p$-group with $\mathsf{D}\left(H\right)\le p^n$. Let $G=H\oplus C_{m{p}^n}$. For any integer $k\ge 2$, we prove that ${\mathsf{s}}_{km{p}^n}\left(G\right)=(k+1)m{p}^n+\mathsf{D}\left(H\right)?2=km{p}^n+\mathsf{D}\left(G\right)?1.$ This verifies the above conjecture in this case. We also provide asymptotically tight bounds for zero-sum invariants $\mathsf{D}\left(G\right)$, ${\mathsf{s}}_{kn}\left(G\right)$ and $\eta \left(G\right)$ for a class of abelian groups with large exponent.     

Valuations and rank of ordered abelian groups

It is shown that there exists an ordered abelian group that has no smallest positive element and that has no sequence of nonzero elements converging to zero. Some formulae for the rank of ordered abelian groups have been derived and a necessary condition for an order type to be rank of an ordered abelian group has been discussed. These facts have been translated to the spectrum of a valuation ring using some well-known results in valuation theory.

     

A note on Selmer groups of abelian varieties over the trivializing extensions

We prove that for any abelian variety defined over a number field that is not isogenous to a product of CM elliptic curves, the pontrjagin dual of the Selmer group of the abelian variety over the trivializing extension has no nonzero pseudo-null submodules.

     

On orthogonal orthomorphisms of cyclic and non‐abelian groups. II

We construct sets of three pairwise orthogonal orthomorphisms of Z_3n, n not divisible by either 2 or 3, n ≠ 7, 17. Combined with results in the literature, this reduces the problem of determining for which v, there exist three pairwise orthogonal orthomorphisms of Z_v to the case v = 9p, p > 3 a prime. This yields new lower bounds for the number of pairwise orthogonal orthomorphisms of classes of dihedral groups of doubly even order, and classes of linear groups. These results also find application in the construction of Z‐cyclic triplewhist tournaments. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 195–209, 2007     

A classification of the automorphism groups of polarized abelian threefolds over finite fields

We give a classification of maximal elements of the set of finite groups that can be realized as the automorphism groups of polarized abelian threefolds over finite fields.     

On a classification of the automorphism groups of polarized abelian surfaces over finite fields

We give a classification of maximal elements of the set of finite groups that can be realized as the full automorphism groups of polarized abelian surfaces over finite fields.     

Extensions of partially ordered partial abelian monoids

The notion of a partially ordered partial abelian monoid is introduced and extensions of partially ordered abelian monoids by partially ordered abelian groups are studied. Conditions for the extensions to exist are found. The cases when both the above mentioned structures have the Riesz decomposition property, or are lattice ordered, are treated. Some applications to effect algebras and MV-algebras are shown.