College Algebra Exam Review 74

College Algebra Exam Review 74 - G be a group and suppose e...

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Chapter 2 Basic Theory of Groups 2.1. First Results In the previous chapter, we saw many examples of groups and finally ar- rived at a definition, or collection of axioms, for groups. In this section we will try our hand at obtaining some first theorems about groups. For many students, this will be the first experience with constructing proofs concerning an algebraic object described by axioms. I would like to urge both students and instructors to take time with this material and not to go on before mastering it. Our first results concern the uniqueness of the identity element in a group. Proposition 2.1.1. (Uniqueness of the identity). Let
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Unformatted text preview: G be a group and suppose e and e are both identity elements in G ; that is, for all g 2 G , eg D ge D e g D ge D g . Then e D e . Proof. Since e is an identity element, we have e D ee . And since e is an identity element, we have ee D e . Putting these two equations together gives e D e . n Likewise, inverses in a group are unique : Proposition 2.1.2. (Uniqueness of inverses). Let G be a group and h;g 2 G . If hg D e , then h D g 1 . Likewise, if gh D e , then h D g 1 . Proof. Assume hg D e . Then h D he D h.gg 1 / D .hg/g 1 D eg 1 D g 1 . The proof when gh D e is similar. n 84...
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