Counting subgroups of the groups ${\Bbb Z}_{n_1} \times \cdots \times {\Bbb Z}_{n_k}$: a survey

Abstract

We present a survey of exact and asymptotic formulas on the number of cyclic subgroups and total number of subgroups of the groups ${\Bbb Z}_{n_1} \times \cdots \times {\Bbb Z}_{n_k}$, where $k\ge 2$ and $n_1,\ldots,n_k$ are arbitrary positive integers.