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 $-1$).