A group is a bounded subset of another group

Question

What does it mean when people say that a group is a bounded subset of another group?

I need decide whether $SO_2(\mathbb C)$ is a bounded subset of $\mathbb C^{2\times 2}$. What definition should I use? What is the "distance" function?

Answer 1

Take any norm on $\mathbb C^{2\times2}$ (for instance $\left\lVert\left[\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right]\right\rVert=\sqrt{\lvert a\rvert^2+\lvert b\rvert^2+\lvert c\rvert^2+\lvert d\rvert^2}$) and use the distance $d(M,N)=\lVert M-N\rVert$.