College Algebra Exam Review 107

College Algebra Exam Review 107 - G . 2.4.5. For any...

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2.4. HOMOMORPHISMS AND ISOMORPHISMS 117 Corollary 2.4.24. A k –cycle is even if k is odd and odd if k is even. Proof. According to Exercise 1.5.5 , a k cycle can be written as a product of .k ± 1/ 2–cycles. n Exercises 2.4 2.4.1. Let ' be the map from symmetries of the square into S 4 induced by the numbering of the vertices of the square in Figure 2.4.1 on page 111 . Complete the tabulation of '.±/ for ± in the symmetry group of the square, and verify the homomorphism property of ' by computation. 2.4.2. Let be the map from the symmetry group of the square into S 2 induced by the labeling of the diagonals of the square as in Figure 2.4.2 on page 112 . Complete the tabulation of and verify the homomorphism property by computation. Identify the kernel of . 2.4.3. Prove Proposition 2.4.10 . 2.4.4. Prove part (b) of Proposition 2.4.12 . Note: Do not assume that ' has an inverse function ' ± 1 W H ±!
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Unformatted text preview: G . 2.4.5. For any subgroup A of a group G , and g 2 G , show that gAg 1 is a subgroup of G . 2.4.6. Show that every subgroup of an abelian group is normal. 2.4.7. Let ' W G ! H be a homomorphism of groups with kernel N . For a;x 2 G , show that '.a/ D '.x/ , a 1 x 2 N , aN D xN: Here aN denotes f an W n 2 N g . 2.4.8. Let ' W G ! H be a homomorphism of G onto H . If A is a normal subgroup of G , show that '.A/ is a normal subgroup of H . 2.4.9. Dene a map from D n to C 2 D f 1 g by ./ D 1 if does not interchange top and bottom of the n-gon, and ./ D 1 if does interchange top and bottom of the n-gon. Show that is a homomorphism. The following exercises examine an important homomorphism from S n to C 2 D f 1 g (for any n )....
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