= Center quotient of the real Heisenberg group
{title2=$H/Z(H)\cong(\mathbb R^2,+)$}
In coordinates $h(x,y,z)=\left(\begin{smallmatrix}1&x&y\\0&1&z\\0&0&1\end{smallmatrix}\right)$, the <Heisenberg group> law is $h(x,y,z)h(a,b,c)=h(x+a,y+b+xc,z+c)$. Commutation for every $(a,b,c)$ forces $xc=az$, hence $x=z=0$. The <center of a group> is therefore the $y$-axis. The surjective <group homomorphism> $h(x,y,z)\mapsto(x,z)$ has exactly this <kernel of a group homomorphism>, giving the stated <quotient group> by the <first isomorphism theorem>.
Back to article page
