  The following problem has been known since the 80's. Let $\Gamma$ be an
Abelian group of order $m$ (denoted $|\Gamma|=m$), and let $t$ and $m_i$, $1
\leq i \leq t$, be positive integers such that $\sum_{i=1}^t m_i=m-1$.
Determine when $\Gamma^*=\Gamma\setminus\{0\}$, the set of non-zero elements of
$\Gamma$, can be partitioned into disjoint subsets $S_i$, $1 \leq i \leq t$,
such that $|S_i|=m_i$ and $\sum_{s\in S_i}s=0$ for every $i$, $1 \leq i \leq
t$. It is easy to check that $m_i\geq 2$ (for every $i$, $1 \leq i \leq t$) and
$|I(\Gamma)|\neq 1$ are necessary conditions for the existence of such
partitions, where $I(\Gamma)$ is the set of involutions of $\Gamma$. It was
proved that the condition $m_i\geq 2$ is sufficient if and only if
$|I(\Gamma)|\in\{0,3\}$. For other groups (i.e., for which $|I(\Gamma)|\neq 3$
and $|I(\Gamma)|>1$), only the case of any group $\Gamma$ with
$\Gamma\cong(Z_2)^n$ for some positive integer $n$ has been analyzed completely
so far, and it was shown independently by several authors that $m_i\geq 3$ is
sufficient in this case. Moreover, recently Cichacz and Tuza proved that, if
$|\Gamma|$ is large enough and $|I(\Gamma)|>1$, then $m_i\geq 4$ is sufficient.
In this paper we generalize this result for every Abelian group of order $2^n$.
Namely, we show that the condition $m_i\geq 3$ is sufficient for $\Gamma$ such
that $|I(\Gamma)|>1$ and $|\Gamma|=2^n$, for every positive integer $n$. We
also present some applications of this result to graph magic- and
anti-magic-type labelings.
