Subgroup_of_Infinite_Cyclic_Group_is_Infinite_Cyclic_Group

Let $G = \gen a$ be an infinite cyclic group generated by $a$, whose identity is $e$. Let $g \in G, g \ne e: \exists k \in \Z, k \ne 0: g = a^k$. Let $H = \gen g$. Then $H \le G$ and $H \cong G$. Thus, all non-trivial subgroups of an infinite cyclic group are themselves infinite cyclic groups. A subgroup of $G = \gen a$ is denoted as follows $n G := \gen {a^n}$ This notation is usually used in the context of $\struct {\Z, +}$, where $n \Z$ is (informally) understood as '''the set of integer multiples of $n$'''.