Intersections of pure subgroups in abelian groups

New constructions of strongly regular Cayley graphs on abelian non p-groups

In this paper the stability of the quadratic equation is considered on arbitrary groups. Since the quadratic equation is stable on Abelian groups, this paper examines the stability of the quadratic equation on noncommutative groups. It is shown that the quadratic equation is stable on $n$-Abelian groups when $n$ is a positive integer. The stability of the quadratic equation is also established on the noncommutative group $T(2, K)$, where $K$ is an arbitrary commutative field. It is proved that every group can be embedded into a group in which the quadratic equation is stable.

Given an abelian group G of order n, and a finite non-empty subset A of integers, the Davenport constant of G with weightA, denoted by DA(G), is defined to be the least positive integer t such that, for every sequence (x1,..., xt) with xi ∈ G, there exists a non-empty subsequence \((x_{j_1},\ldots, x_{j_l})\) and ai ∈ A such that \(\sum_{i=1}^{l}a_ix_{j_i} = 0\). Similarly, for an abelian group G of order n, EA(G) is defined to be the least positive integer t such that every sequence over G of length t contains a subsequence \((x_{j_1} ,\ldots, x_{j_n})\) such that \(\sum_{i=1}^{n}a_ix_{j_i} = 0\), for some ai ∈ A. When G is of order n, one considers A to be a non-empty subset of {1,..., n − 1 }. If G is the cyclic group \({\Bbb Z}/n{\Bbb Z}\), we denote EA(G) and DA(G) by EA(n) and DA(n) respectively.

Minimal sumsets in finite solvable groups

On diagrams for abelian groups

All groups herein are assumed to be abelian. It was not until the 1940's that it was known that a subgroup of an infinite direct sum of finite cyclic groups is again a direct sum of cyclics. This result rests on a general criterion due to Kulikov [7] for a primary abelian group to be a direct sum of cyclic groups. If G is p-primary, Kulikov's criterion presupposes that G has no elements (other than zero) having infinite p-height. For such a group G, the criterion is simply that G be the union of an ascending sequence of subgroups Hn where the heights of the elements of Hn computed in G are bounded by some positive integer λ(n). The theory of abelian groups has now developed to the point that totally projective groups currently play much the same role, at least in the theory of torsion groups, that direct sums of cyclic groups and countable groups played in combination prior to the discovery of totally projective groups and their structure beginning with a paper by R. Nunke [11] in 1967.

Let $S$ be a subsemigroup which contains 0 of a torsion-free abelian (additive) group. Then $S$ is called a grading monoid (or a $g$-monoid). The group $ \{s-s'|s,s'\in S\}$ is called the quotient group of $S$, and is denored by $q(S)$. Let $R$ be a commutative ring. The total quotient ring of $R$ is denoted by $q(R)$. Throught the paper, we assume that a $g$-monoid properly contains $ \{0\}$. A commutative ring is called a ring, and a non-zero-divisor of a ring is called a regular element of the ring. We consider integral elements over the semigroup ring $ R[X;S]$ of $S$ over $R$. Let $S$ be a $g$-monoid with quotient group $G$. If $ n\alpha\in S$ for an element $ \alpha$ of $G$ and a natural number $n$ implies $ \alpha\in S$, then $S$ is called an integrally closed semigroup. We know the following fact: ${\bf Theorem~1}$ ([G2, Corollary 12.11]). Let $D$ be an integral domain and $S$ a $g$-monoid. Then $D[X;S]$ is integrally closed if and only if $D$ is an integrally closed domain and $S$ is an integrally closed semigroup. Let $R$ be a ring. In this paper, we show that conditions for $R[X;S]$ to be integrally closed reduce to conditions for the polynomial ring of an indeterminate over a reduced total quotient ring to be integrally closed (Theorem 15). Clearly the quotient field of an integral domain is a von Neumann regular ring. Assume that $q(R)$ is a von Neumann regular ring. We show that $R[X;S]$ is integrally closed if and only if $R$ is integrally closed and $S$ is integrally closed (Theorem 20). Let $G$ be a $g$-monoid which is a group. If $R$ is a subring of the ring $T$ which is integrally closed in $T$, we show that $R[X;G]$ is integrally closed in $T[X;S]$ (Theorem 13). Finally, let $S$ be sub-$g$-monoid of a totally ordered abelian group. Let $R$ be a subring of the ring $T$ which is integrally closed in $T$. If $g$ and $h$ are elements of $T[X;S]$ with $h$ monic and $gh\in R[X;S]$, we show that $g\in R[X;S]$ (Theorem 24).

Let n be a positive integer, Bn the closed unit ball in Euclidean n-space, and X any countable CW-complex of dimension at most n. It is shown that the free Abelian topological group on Bn, F(Bn), has F(X) as a closed subgroup. It is also shown that for every differentiable manifold Y of dimension at most n, F(Y) is a closed subgroup of F(Bn).

Let k be a number field, let A be a commutative algebraic group defined over k and let n be a positive integer. We will prove that for almost all primes p ∈ k, the existence of a counterexample to the localglobal divisibility by p in A assures the existence of a counterexample to the local-global divisibility by p in A, for all positive integers s. In particular, we will prove the existence of counterexamples to the localglobal divisibility by 2 and 3, for every n ≥ 2, in the case when k = Q and A is an elliptic curve. Furthermore, we will show a method to find numerical examples. We will use this method to produce numerical counterexamples to the local-global divisibility by 2, for every n ≥ 2, in elliptic curves defined over Q.

A problem of Călugăreanu, Chekhlov and Krylov asks which abelian groups have the property that all of their pure subgroups that are endomorphic images are necessarily summands. Complete answers are given for the groups in certain classes (for example, the torsion groups). On the other hand, examples are constructed that show a complete solution to the general problem is likely to be quite difficult.

In this chapter, we are going to discuss a basic concept: pure subgroup. This concept has been one of the most fertile notions in the theory since its inception in a paper by the pioneer H. Prufer. The relevance of purity in abelian group theory, and later in module theory, has tremendously grown with time. While abelian groups have been major motivation for a number of theorems in category theory, purity has served as a prototype for relative homological algebra, and has played a significant role in model theory as well.

In the work of Herden and Strüngmann (In Models, modules and Abelian groups (2008) 169–186 de Gruyter), an embedding problem for torsion-free Abelian groups was considered. It was shown for a large class of such groups, including the class of all bounded extensions of completely decomposable groups, that any member of the class can be purely embedded into some completely decomposable group. Moreover, an algorithm was given that determines explicitly the pure embedding and the completely decomposable overgroup. We continue the approach from the work of Herden and Strüngmann (In Models, modules and Abelian groups (2008) 169–186 de Gruyter) improving the algorithm and extending the main theorem to a broader class of torsion-free Abelian groups including some Hawaiian groups from the article of Mader and Strüngmann (J. Algebra 229 (2000) 205–233) and thus complementing the main result from the article of Strüngmann (Proc. Amer. Math. Soc. 137 (2009) 3657–3668). A byproduct and starting point for this generalization will be a discussion of the following group class problem: Which groups $G$ have the property that for any cardinal $\kappa$ any subgroup $U$ of the direct sum $G^{(\kappa)}$ is the kernel of some endomorphism of $G^{(\kappa)}$?

An extremal problem on bigraphic pairs with an [formula omitted]-connected realization

We consider redundant binary joint digital expansions of integer vectors. The redundancy is used to minimize the Hamming weight,i.e., the number of non-zero digit vectors. This leads to efficient linear combination algorithms in abelian groups, which are used in elliptic curve cryptography, for instance.If the digit set is a set of contiguous integers containing zero, a special syntactical condition is known to minimize the weight. We analyse the optimal weight of all non-negative integer vectors with maximum entry less thanN. The expectation and the variance are given with a main term and a periodic fluctuation in the second-order term. Finally, we prove asymptotic normality.

We define the positive diameter of a finite group G with respect to a generating set A ⊆ G to be the smallest non-negative integer n such that every element of G can be written as a product of at most n elements of A. This invariant, which we denote by , can be interpreted as the diameter of the Cayley digraph induced by A on G.