The cross number of finite abelian groups. II (Q1332361)

[For part I cf. the first author, J. Number Theory 48, 219-223 (1994; Zbl 0814.20033).] Let \(G\) be an additively written finite abelian group, \(G=C_{n_1}\oplus\dots\oplus C_{n_r}\) its direct decomposition into cyclic groups of prime power order, and \(\text{exp}(G)\) its exponent. Set \(k^*(G)=\sum^r_{i=1} (n_i-1)/n_i\) and \(K^*(G)=1+\text{exp}(G) k^* (G)\). Let \(S=(g_1,\dots, g_l)\) be a sequence in \(G\). \(S\) is called a zero sequence if \(\sum^l_{i=1} g_i=0\). Set \(k(S)=\sum^l_{i=1} 1/\text{ord} (g_i)\). Let \({\mathcal U} (G)\) denote the set of minimal zero sequences in \(G\). Set \(K(G)=\text{exp} (G)\max\{k(S)\mid S\in {\mathcal U}(G)\}\). It is known that \(K^*(G)\leq K(G)\), and so far there is known no finite abelian group \(G\) with \(K^*(G)<K(G)\). However, \(K^*(G)=K(G)\) has been proved only for very special types of cyclic groups and for \(p\)-groups. In this paper, the authors tackle the problem of determining \(K(G)\) with a new method, and hence \(K^*(G)=K(G)\) can be proved for further classes of finite groups. The main results are the following theorems: Theorem 1. Let \(G=C_{p^m}\oplus C_{p^n}\oplus C_{q^s}\) with distinct primes \(p\), \(q\) and \(m, n, s\in\mathbb{N}_+\). Then \(K(G)=K^*(G)\). Theorem 2. Let \(G=\bigoplus^r_{i=1} C_{p_i^{n_i}}\oplus C_{q^s}\) with distinct primes \(p_1,\dots, p_r\), \(q\) integers \(n_1,\dots, n_r\in\mathbb{N}_+\) and \(s\in\mathbb{N}\). Then \(K(G)=K^*(G)\) in the following two cases: (a) \(r\leq 3\) and \(p_1\dots p_r\neq 30\); (b) \(p_k\geq k^3\) for every \(1\leq k\leq r\).