Zero-sum partitions of Abelian groups of order 2n

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 mi, 1 <= i <= t, be positive integers such that sigma(t)(i)=1 m(i) = m - 1. Determine when gamma(& lowast;) = gamma \ {0}, the set of non-zero elements of gamma, can be partitioned into disjoint subsets S-i, 1 <= i <= t, such that |S-i| = m(i) and sigma(s is an element of Si) s = 0 for every i is an element of [1, t].It is easy to check that m(i) >= 2 (for every i is an element of [1, t]) and |I(gamma)| &NOTEQUexpressionL; 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 )>= 2 is sufficient if and only if |I(gamma)| is an element of {0, 3} (see Zeng, (2015)).For other groups (i.e., for which |I(gamma)&NOTEQUexpressionL; 3 and |I(gamma)| > 1), only the case of any group gamma with gamma expressionpproximexpressiontely equexpressionl to (Z(2))(n) for some positive integer n has been analyzed completely so far, and it was shown independently by several authors that mi > 3 is sufficient in this case. Moreover, recently Cichacz and Tuza (2021) proved that, if |gamma| is large enough and |I(gamma)|> 1, then m(i) >= 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) >= 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.