Is $O_n$ isomorphic to $SO_n \times \{\pm I\}$?

Question

This question is taken directly from Artin's "Algebra", on page 150:

Is $O_{n}$ isomorphic to the product group of $SO_{n} \times \{\pm1\}$?

Here, $O_{n}$ is defined as the group of orthogonal matrices, $SO_{n}$ is the special group of orthogonal matrices (i.e. orthogonal matrices which have determinant or $1$ or $-1$).

Answer 1

No. $A\in O_n$ iff the columns of $A$ form an orthonormal system, i.e. $A^TA=I$, see e.g.wiki, hence $\det A=\pm 1$ already there. And $SO_n=\{A\in O_n\mid \det A=1\}$.

Well, if $n$ is even, $\det(-I)=1$, so $A\mapsto (-I)\cdot A$ is invariant in $O_n$..

Answer 2

For it to be a direct product we need an element of order 2 with determinant -1. Thus the eigenvalues are all $\pm 1$. The diagonal matrix with a single -1 and the rest 1 has the right property. But this does not commute with $SO_n$. In order for it to commute with $SO_n$ we need a scalar matrix. So $O_n$ is a direct product of $SO_n$ and <-1> iff $n$ is odd.

Answer 3

When $n$ is odd it is clear we have $O_n \cong SO_n \times \{\pm I\}$ via the product map. Now if $n$ is even, the center of $O_n$ is $\pm I$ while the center of $SO_n$ is also $\pm I$. If $O_n \cong SO_n \times \pm I$ when $n$ is even, they would have isomorphic centers. But then

$$\pm I \not\cong \pm I \times \pm I$$

which is a contradiction. So for $n$ even, $O_n$ is not isomorphic to $SO_n \times \pm I$.