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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 $-1$).

  • 0
    What property of a product are you having trouble verifying (or disproving)?2012-10-01
  • 0
    compare also http://math.stackexchange.com/questions/29279/why-is-the-orthogonal-group-operatornameo2n-mathbb-r-not-the-direct-prod?rq=12012-12-06

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