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College Algebra Exam Review 122

College Algebra Exam Review 122 - Definition 2.6.16 Let a...

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132 2. BASIC THEORY OF GROUPS Let us look again at our main example, the equivalence relation on a group G determined by a subgroup H , whose equivalence classes are the left cosets of H in G . We would like to define a canonical surjective map on G whose fibers are the left cosets of H in G . Definition 2.6.14. The set of left cosets of H in G is denoted G=H . The surjective map W G ! G=H defined by .a/ D aH is called the canonical projection or quotient map of G onto G=H . Proposition 2.6.15. The fibers of the canonical projection W G ! G=H are the left cosets of H in G . The equivalence relation on G deter- mined by is the equivalence relation H . Proof. We have 1 .aH/ D f b 2 G W bH D aH g D aH . Furthermore, a b , aH D bH , a H b . n Conjugacy We close this section by introducing another equivalence relation that is extremely useful for studying the structure of groups:
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Unformatted text preview: Definition 2.6.16. Let a and b be elements of a group G . We say that b is conjugate to a if there is a g 2 G such that b D gag ± 1 . You are asked to show in the Exercises that conjugacy is an equiva-lence relation and to find all the conjugacy equivalence classes in several groups of small order. Definition 2.6.17. The equivalence classes for conjugacy are called con-jugacy classes. Note that the center of a group is related to the notion of conjugacy in the following way: The center consists of all elements whose conjugacy class is a singleton. That is, g 2 Z.G/ , the conjugacy class of g is f g g ....
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