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    I need to find a surjective homomorphism from O3 onto SO3. The only thing that I can think of is a map which sends A to A^2, but it's not an homomorphism. Any idea? Thanks. D.

    This should work if I've understood the notation (and not forgot some of algebra).
     \phi : O(3) \to SO(3) defined by  \phi(A) = A if  \det A = 1 and  \phi(A) = -A if  \det A = -1.

    It's surjective since every A in SO(3) is also in O(3) and has determinant 1, so it's just mapped to itself. Checking it's a homomorphism can be done by considering the cases for determinant of A and B.

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