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College Algebra Exam Review 123

College Algebra Exam Review 123 - and O for the coset O...

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2.7. QUOTIENT GROUPS AND HOMOMORPHISM THEOREMS 133 Exercises 2.6 2.6.1. Consider any surjective map f from a set X onto another set Y . We can define a relation on X by x 1 x 2 if f .x 1 / D f .x 2 / . Check that this is an equivalence relation. Show that the associated partition of X is the partition into “fibers” f 1 .y/ for y 2 Y . The next several exercises concern conjugacy classes in a group. 2.6.2. Show that conjugacy of group elements is an equivalence relation. 2.6.3. What are the conjugacy classes in S 3 ? 2.6.4. What are the conjugacy classes in the symmetry group of the square D 4 ? 2.6.5. What are the conjugacy classes in the dihedral group D 5 ? 2.6.6. Show that a subgroup is normal if, and only if, it is a union of conjugacy classes. 2.7. Quotient Groups and Homomorphism Theorems Consider the permutation group S n with its normal subgroup of even per- mutations. For the moment write E for the subgroup of even permutations
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Unformatted text preview: and O for the coset O D .12/ E D E .12/ consisting of odd permuta-tions. The subgroup E is the kernel of the sign homomorphism ± W S n ²! f 1; ² 1 g . Since the product of two permutations is even if, and only if, both are even or both are odd, we have the following multiplication table for the two cosets of E : E O E E O O O E The products are taken in the sense mentioned previously; namely the product of two even permutations or two odd permutations is even, and the product of an even permutation with an odd permutation is odd. Thus the multiplication on the cosets of E reproduces the multiplication on the group f 1; ² 1 g . This is a general phenomenon: If N is a normal subgroup of a group G , then the set G=N of left cosets of a N in G has the structure of a group....
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