# What is the group O(4)/H, where H is the center of O(4)?

What is the group $O(4)/H$?

Here $O(4)$ is the group of orthogonal matrices and H is the center of $O(4)$.

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It's the group of inner automorphisms of $O(4)$. –  David Roberts Feb 26 '11 at 4:01
This looks a bit like homework to me (mathoverflow.net/faq#whatnot). What is your motivation for considering this question? –  David Roberts Feb 26 '11 at 4:04
Someone can make a nice question out of nice applications of this post! –  Romeo Feb 27 '11 at 2:00
If this answer is sufficient for your purposes, why not accept it? –  Yemon Choi Oct 2 '11 at 1:58

This group is isomorphic to $(SO(3) \times SO(3)) \rtimes \{\pm 1\}$, as is discussed in this question and its comments. (The order two factor comes from the fact that $O(4) = SO(4) \rtimes \{\pm 1\},$ and that $H \subset SO(4)$.) [Added as per Scott Carnahan's comment below: The order two factor shouldn't be a direct factor, but a semi-direct factor. I leave it as an exercise to determine its action on $SO(3) \times SO(3)$. If someone reading this knows the correct LaTeX command for a semi-direct product, feel free to make the appropriate edit.]
Is it really a group-theoretic direct product? This seems like a situation where one should use the semidirect product symbol $\rtimes$ (and possibly specify the involution). –  S. Carnahan Feb 26 '11 at 7:08