Integer shear group ($B_n=\left(\begin{array}{cc}1& n\\ 0& 1\end{array}\right),B_n{B}_m=B_{n+m}$)

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 $(x+ny,y)$.