Let $O_n(\mathbb{R})$ be the group of real orthogonal $n\times n$ matrices and $SO_n(\mathbb{R})$ be the group of real orthogonal matrices with determinant $1$.
(i) Show that $O_n(\mathbb{R}) = SO_n(\mathbb{R}) × \{\pm I_n\}$ if and only if $n$ is odd.
(ii) Show that if $n$ is even, then $O_n(\mathbb{R})$ is not the direct product of $SO_n(\mathbb{R})$ with any
normal subgroup.
Here is the progress I have made so far:
(i) If $n$ is odd then consider the map $\phi: SO_n(\mathbb{R}) × \{\pm I_n\} \to O_n : (A,B) \to AB$. This is a homomorphism as all the elements of $\{\pm I_n\}$ commute with the elements of $SO_n(\mathbb{R})$. Furthermore it is injective as is $AB = CD$ then since $A,C$ have determinant $1$ we get that $B,D$ have the same determinant so $B,D$ are the same matrix so $A,C$ are the same as well. It is also surjective as if $E \in O_n(\mathbb{r})$ then either $E$ or $-E \in SO_n(\mathbb{R})$ and so either $(E,I_n)$ or $(-E,-I_n)$ maps to $E$. Hence we have an isomorphism and they are the same.
If $n$ is even then note that $O_n(\mathbb{R})$ has center of order $2$ while $ SO_n(\mathbb{R}) × \{\pm I_n\}$ has center of order $4$ so they are not isomorphic.
(ii) I am having trouble with this bit. I can't even manage to show $O_n(\mathbb{R})$ is not isomorphic to $SO_n(\mathbb{R})$ for even $n$.
Any help is much appreciated.
Best Answer
For the last part, let $H$ be a subgroup(necessarily normal) such that $O_n=SO_n\odot H$. We then have $H\cap SO_n=\{I_n\}$.
Let $h\in H$ which is not $I_n$, so $h\notin SO_n$. Since $SO_n$ has index $2$ in $O_n$, $O_n$ is generated by $h$ and elements of $SO_n$.
Now $h$ commutes with every element of $SO_n$ (direct product properties), and commutes with itself, so $h$ commutes with any element of $O_n$. Hence $h=- I_n$. But since $n$ is even, $h\in SO_n$, a contradiction.