Can anyone suggest a reference to an explicit formula giving, in the standard matrix notation, a homomorphism from SO(3,q) to PGL(2,q) (classical matrix groups: orthogonal of dimension 3 and projective general linear of dimension 2 over finite field of order q. To write the reverse map, from PGL(2,q) to SO(3,q) is easy: this is just the adjoint representation of GL(2,q).
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Why isn't $M \mapsto I$ sufficient, i.e., send every element of $SO(3)$ to the identity element of $PGL(2, q)$? That's a homomorphism. – John Hughes Oct 28 '14 at 18:03
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I don't have the book "The geometry of the classical groups" by Donald E. Taylor at hand now, but according to google books Theorem 11.6 claims this isomorphism (but I cannot see the proof). – j.p. Oct 29 '14 at 14:52