Hey again, I need a lot of help here.

Is $\displaystyle O_n$ isomorphic to the product group $\displaystyle SO_n \times \{\pm I \}$?

I can show that for $\displaystyle n=2$, then No it is not, and for $\displaystyle n=3$ it is isomorphic but how do I show it in general.

Thank for your help.