Counting subgroups of the groups \({\mathbb{Z}}_{n_1} \times \cdots \times{\mathbb{Z}}_{n_k}\): a survey (Q6610540)

It is well known that the problem of counting the subgroups of finite abelian groups reduces to \(p\)-groups. The author provides a detailed review of the results obtained earlier for \(p\)-groups by various authors including [\textit{G. Bhowmik}, Acta Arith. 74, No. 2, 155--159 (1996; Zbl 0848.15015); \textit{S. Delsarte}, Ann. Math. (2) 49, 600--609 (1948; Zbl 0031.34102); \textit{G. A. Miller}, Ann. Math. (2) 6, 1--6 (1904; JFM 35.0162.03); Proc. Natl. Acad. Sci. USA 25, 258--262 (1939; JFM 65.0065.02); \textit{V. N. Sokuev}, Math. Notes 12, 774--778 (1973; Zbl 0271.20009); \textit{Y. Yeh}, Bull. Am. Math. Soc. 54, 323--327 (1948; Zbl 0033.15102)]. Instead of \(p\)-groups the author considers the groups of the form \(Z_{n_1}\times \cdots \times Z_{n_k}\) where \(n_1,\ldots , n_k\) are arbitrary positive integers. If \(k=2\), \(k=3\) or \(k=4\) then the exact and asymptotic formulas for the total number of subgroups and the number of its cyclic subgroups are presented.\N\NFor the entire collection see [Zbl 1539.11005].