On $n$-Sum Groups.

n-cyclicizer groups

on $n$-kappe groups

let $g$ be an infinite group and $nin {3‎, ‎6}cup{2^k| kin mathbb{n}}$‎. ‎in this paper‎, ‎we prove that $g$ is an $n$-kappe group if and only if for any two infinite subsets $x$ and $y$ of $g$‎, ‎there exist $xin x$ and $yin y$ such that $[x^n‎, ‎y‎, ‎y]=1$‎.

On Zero-sum Partitions of Abelian Groups

In this paper, confirming a conjecture of Kaplan et al., we prove that every abelian group G, which is of odd order or contains exactly three involutions, has the zerosum-partition property. As a corollary, every tree with |G| vertices and at most one vertex of degree 2 is G-anti-magic.

Some properties of n-capable and n-perfect groups

In this article we introduce the notion of n-capable groups. It is shown that every group G admits a uniquely determined subgroup (〖Z^n)〗^* (G) which is a characteristic subgroup and lies in the n-centre subgroup of the group G. This is the smallest subgroup of G whose factor group is n-capable. Moreover, some properties of n-central extension will be studied.

On short zero-sum subsequences over p-groups

Let G be a finite abelian group with exponent n. Let s(G) denote the smallest integer l such that every sequence over G of length at least l has a zero-sum subsequence of length n. For p-groups whose exponent is odd and sufficiently large (relative to Davenport’s constant of the group) we obtain an improved upper bound on s(G), which allows to determine s(G) precisely in special cases. Our resu...