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On Davenport’s constant of finite Abelian groups with rank three. (English) Zbl 0971.20032
The Davenport constant $$D(G)$$ of a finite Abelian group $$G$$ is defined as the smallest integer $$d\in\mathbb{N}$$ such that every sequence $$S$$ of $$G$$ with length $$|S|\geq d$$ contains a subsequence with sum zero. Here a sequence $$S$$ (in $$G$$) is an element of the free Abelian monoid $${\mathcal F}(G)$$ with basis $$G$$. If $$G=C_{n_1}\oplus\cdots\oplus C_{n_r}$$ with $$1<n_1\mid\cdots\mid n_r$$ then define $$M(G)=1+\sum_{i=1}^r(n_i-1)$$. Then $$M(G)\leq D(G)$$ and the following results are already known: 1. If $$G$$ is either a $$p$$-group or has rank $$\leq 2$$, then $$M(G)=D(G)$$. 2. For every $$r\geq 4$$ there are infinitely many groups $$G$$ with rank $$r$$ such that $$M(G)<D(G)$$.
The paper deals with the following open problem (van Emde Boas): Does $$M(G)=D(G)$$ hold for all the finite Abelian groups with rank 3? Using another invariant $$\nu(G)$$ (dealing with the structure of $$\Sigma(S)$$ – the set of sums of non-empty subsequences of $$S$$ – for long zerofree sequences $$S$$), the class of the groups such that $$D(G)=\nu(G)+2$$ is enlarged with groups of rank two.
The main result is Theorem 5.2. Let $$H=C_{n_1}\oplus C_{n_2}$$ with $$1<n_1\mid n_2$$ and $$D(H)=\nu(H)+2$$. Let $$m\in\mathbb{N}$$ be a positive integer such that every odd prime divisor $$p$$ of $$m$$ satisfies Property D (if every sequence $$S=a^{n-1}b^{n-1}\prod_{i=1}^{n-1}c_i\in{\mathcal F}(C_n\oplus C_n)$$, which contains no short zero subsequence, satisfies $$c_1=\cdots=c_{n-1}$$) and $$(p-4)n_1-n_2+3\geq (p-1)^2(p^2-2)$$. Then $$D(G)=\nu(G)+2$$ for $$G=C_{n_1m}\oplus C_{n_2m}$$.

MSC:
 20K01 Finite abelian groups 11B83 Special sequences and polynomials 20D60 Arithmetic and combinatorial problems involving abstract finite groups
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