Payton Rasmussen

2022-10-09

(1) Prove that O(n) is homeomorphic to $\mathrm{S}\mathrm{O}\left(n\right)×{\mathbb{Z}}_{2}$. (2) Are these two isomorphic as topological groups?
(O(n) : Orthogonal group, SO(n) : Special orthogonal group)
My attempt for (1) :Let's pick any element C from $\mathrm{O}\left(n\right)-\mathrm{S}\mathrm{O}\left(n\right)$, and let
$\begin{array}{rl}& f:\mathrm{S}\mathrm{O}\left(n\right)\to \mathrm{S}\mathrm{O}\left(n\right);\phantom{\rule{thickmathspace}{0ex}}f\left(A\right)=A\\ & g:\mathrm{O}\left(n\right)-\mathrm{S}\mathrm{O}\left(n\right)\to \mathrm{S}\mathrm{O}\left(n\right);\phantom{\rule{thickmathspace}{0ex}}g\left(A\right)=CA\end{array}$
Then f and g are homeomorphisms. Since SO(n) is compact, ${g}^{-1}\left(\mathrm{S}\mathrm{O}\left(n\right)\right)=\mathrm{O}\left(n\right)-\mathrm{S}\mathrm{O}\left(n\right)$ is also compact. So, they are closed in the subspace topology on O(n). Thus, by the glueing lemma, the function

becomes continuous. We can easily confirm that $\phi$ is bijective and ${\phi }^{-1}$ is continuous.
Questions:
(1) I'm not sure I'm going in the right direction (in proving (1)). I used glueing lemma to justifying continuousness of the function $\phi$, but I doubt whether I'm using that lemma in the right place.
(2) I tried to prove that two groups are NOT isomorphic, but I can't find any key for that. I tried to compare orders of elements in two groups, but I couldn't find some good elements to compare... Is there any simple way to check whether two groups are isomorphic?

antidootnw

Step 1
Consider the mapping $f:O\left(n\right)\to SO\left(n\right)×{\mathbb{Z}}_{2}$ given by
$K\to \left\{\text{det}\left(M\right)\cdot M,\text{det}M\right\}$
This is a well defined one to one and onto map. You can check that the map above is continuous.
Step 2
Now since we know that O(n) is compact and $SO\left(n\right)×{\mathbb{Z}}_{2}$ is Hausdorff , the following map is a homeomorphism.

Riya Andrews

Step 1
Consider the mapping $f:O\left(n\right)\to SO\left(n\right)×{Z}_{2}2$ given by $M\to \left(det\left(M\right)\cdot M,detM\right)$
Step 2
I think your answer is not right. Check det $\left[det\left(M\right)M\right]=\left[det\left(M\right){\right]}^{n+1}$. Thus if n is even, it will be wrong.

Do you have a similar question?