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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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