Infinite cyclic group

An infinite cyclic group is generated by an element of infinite order and is isomorphic to the additive group of integers.

For $G=⟨g⟩$ infinite, every subgroup is $⟨g^m⟩$ for a unique integer $m\ge 0$. This is the classification of subgroups of the additive group of integers transported through a group isomorphism. For $m>0$ the subgroup index is $m$ and the subgroup is again an infinite cyclic group; for $m=0$ it is trivial and has infinite subgroup index.

The integers form an infinite cyclic group under addition, with identity element zero and inverse element $-n$ for $n$. Its two generators are $1$ and $-1$. Every subgroup is $m\mathbb{Z}$ for a unique nonnegative integer $m$: if it is nonzero, take its least positive element and use division with remainder to show that every element is a multiple of it. The zero subgroup corresponds to $m=0$.

These invertible matrices, for $n\in \mathbb{Z}$, form an infinite cyclic group. The displayed multiplication rule, $B_0=I$, and $B_n^{-1}=B_{-n}$ give a group isomorphism from the additive group of integers via $n\mapsto B_n$. Its generator is $B_1$. The subgroups of an infinite cyclic group become $\{B_{mk}:k\in \mathbb{Z}\}$ for integers $m\ge 0$, and their subgroup indices are $m$ when $m>0$. The case $m=0$ is the trivial subgroup and has infinite index. As transformations, these matrices send $(x,y)$ to $(x+ny,y)$.