= Cosets of a coordinate subgroup of the integer Heisenberg group
{title2=$H M(a,b,c)=\{M(a,b,t):t\in\mathbb Z\}$}
Write $M(a,b,c)=\begin{pmatrix}1&a&b\\0&1&c\\0&0&1\end{pmatrix}$ with integer coordinates. Multiplication in this parametrization is $M(a,b,c)M(A,B,C)=M(a+A,b+B+aC,c+C)$. The subgroup $H=\{M(0,0,t):t\in\mathbb Z\}$ has <right cosets> obtained by multiplying on the left by its elements, so its right cosets are exactly the displayed sets with fixed $(a,b)$. In particular, right cosets are indexed by $\mathbb Z^2$. Left cosets differ: $M(a,b,c)H$ fixes $a$ and $b-ac$, since right multiplication changes $b$ by $at$ and $c$ by $t$. This also exhibits why $H$ is not a <normal subgroup> of the <Integer Heisenberg group>.
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