Integer shear group
= Integer shear group
{title2=$B_n=\begin{pmatrix}1&n\\0&1\end{pmatrix},\quad B_nB_m=B_{n+m}$}
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 $m\geq0$, 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 $(x+ny,y)$.