Fundamental Group of the special Euclidean matrix group of the plane How do you do this?  Compute the fundamental group of the special Euclidean group of the plane, that is, all matrices of the form:  $
\left( \begin{array}{ccc}
\cos(z) & \sin(z) & x \\
-\sin(z) & \cos(z)& y \\
0 & 0 & 1\end{array} \right)$
I don't know how to start.  Please give hints.

The upper left $2 \times 2$-block parameterizes a familiar space, call it $X$. The $(x,y)$-entries give a copy of $\mathbb{R}^2$. The group is homeomorphic to $X \times \mathbb{R}^2$ and therefore the fundamental group of the Euclidean group is isomorphic to $\pi_1(X)$.
