College Algebra Exam Review 239

College Algebra Exam Review 239 - automorphism unless r D ....

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5.3. SYMMETRIES OF GROUPS 249 Corollary 5.3.4. Aut . Z p / is cyclic of order p ± 1 . Proof. This follows from the previous result and Corollary 3.6.26 . n The automorphism groups of several other abelian groups are deter- mined in the Exercises. Exercises 5.3 5.3.1. Show that ˛ 7! ˛.Œ1Ł/ is an isomorphism from Aut . Z n / onto the group of units of Z n . 5.3.2. Show that a homomorphism of the additive group Z 2 into itself is determined by a 2-by-2 matrix of integers. Show that the homomorphism is injective if, and only if, the determinant of the matrix is nonzero, and bijective if, and only if, the determinant of the matrix is ˙ 1 . Conclude that the group of automorphisms of Z 2 is isomorphic to the group of 2-by-2 matrices with integer coefficients and determinant equal to ˙ 1 . 5.3.3. Generalize the previous problem to describe the automorphism group of Z n . 5.3.4. Show that any homomorphism of the additive group Q into itself has the form x 7! rx for some r 2 Q . Show that a homomorphism is an
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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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