On Modified Erdős-Ginzburg-Ziv constants of finite abelian groups

Let $G$ be a finite abelian group with exponent $\exp (G)$ and $S$ be a sequence with elements of $G$. We say $S$ is a zero-sum sequence if the sum of the elements in $S$ is the zero element of $G$. For a positive integer $t$, let ${\mathtt{s}}_{t\exp (G)}(G)$ (respectively, ${\mathtt{s}}_{t\exp (G)}^{\prime }(G)$) denote the smallest integer $\ell$ such that every sequence (respectively, zero-sum sequence) $S$ over $G$ with $|S|\ge \ell$ contains a zero-sum subsequence of length $t\exp (G)$. The invariant ${\mathtt{s}}_{t\exp (G)}(G)$ (respectively, ${\mathtt{s}}_{t\exp (G)}^{\prime }(G)$) is called the Generalized Erdős-Ginzburg-Ziv constant (respectively, Modified Erdős-Ginzburg-Ziv constant) of $G$. In this paper, we discuss the relationship between Generalized Erdős-Ginzburg-Ziv constant and Modified Erdős-Ginzburg-Ziv constant, and determine ${\mathtt{s}}_{t\exp (G)}^{\prime }(G)$ for some finite abelian groups.