The Subgroups of Finite Metacyclic Groups

In this paper, the author characterizes the subgroups of a finite metacyclic group $K$ by building a one to one correspondence between certain 3-tuples $(k,l,\beta)\in \mathbb{N}^3$ and all the subgroups of $K$. The results are applied to compute some subgroups of $K$ as well as to study the structure and the number of $p$-subgroups of $K$, where $p$ is a fixed prime number. In addition, the author gets a factorization of $K,$ and then studies the metacyclic $p$-groups, gives a different classification, and describes the characteristic subgroups of a given metacyclic $p$-group when $p\geq3$. A ``reciprocity'' relation on enumeration of subgroups of a metacyclic group is also given.