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Unformatted text preview: automorphism unless r D . Conclude that the automorphism group of Q is isomorphic to Q , namely, the multiplicative group of nonzero rational numbers. 5.3.5. Show that any group homomorphism of the additive group Q 2 is de-termined by rational 2-by-2 matrix. Show that any group homomorphism is actually a linear map, and the group of automorphisms is the same as the group of invertible linear maps. 5.3.6. Think about whether the results of the last two exercises hold if Q is replaced by R . What issue arises? 5.3.7. We can show that Aut . Z 2 Z 2 / S 3 in two ways. (a) One way is to show that any automorphism is determined by an invertible 2-by-2 matrix with entries in Z 2 , that there are six such matrices, and that they form a group isomorphic to S 3 . Work out the details of this approach....
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- Fall '08