College Algebra Exam Review 126

# College Algebra Exam Review 126 - .s Q.t Thus Q is a group...

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136 2. BASIC THEORY OF GROUPS complex numbers of modulus 1, namely T D f z 2 C W j z j D 1 g D f e 2±it W t 2 R g D f e 2±it W 0 ± t < 1 g : So now we have bijections between set R = Z of cosets of R modulo Z , the set Œ0;1/ , and the unit circle T , given by ŒtŁ 7! t ² ŒŒtŁŁ 7! e 2±it D e 2±i.t ± ŒŒtŁŁ/ : Let us write ' for the map t 7! e 2±it from R onto the unit circle, and Q ' for the map ŒtŁ 7! '.t/ D e 2±it . Our discussion shows that Q ' is well deﬁned. We know that the unit circle T is itself a group, and we recall that that the exponential map ' W R ! T is a group homomorphism, namely, '.s C t/ D e 2±i.s C t/ D e 2±is e 2±it D '.s/'.t/: Furthermore, the kernel of ' is precisely Z . We now have a good geometric picture of the quotient group R = Z as a set , but we still have to discuss the group structure of R = Z . The deﬁnition of the product (addition!) on R = Z is ŒtŁ C ŒsŁ D Œt C . But observe that Q '.ŒsŁ C ŒtŁ/ D Q '.Œs C tŁ/ D e 2±i.s C t/ D e 2±is e 2±it D Q
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Unformatted text preview: '.s/ Q '.t/: Thus Q ' is a group isomorphism from the quotient group R = Z to T . Our work can be summarized in the following diagram, in which all of the maps are group homomorphisms, and the map ± is the quotient map from R to R = Z . R ' q q q q q T ± q q q q q ± ± ± ± ± ± q q q q q q q ∼ = Q ' R = Z Example 2.7.5. Recall from Exercise 2.4.20 the “ Ax C b ” group or afﬁne group Aff .n/ consisting of transformations of R n of the form T A; b . x / D A x C b ; where A 2 GL .n; R / and b 2 R n . Let N be the subset consisting of the transformations T E; b , where E is the identity transformation, T E; b . x / D x C b : The composition rule in Aff .n/ is T A; b T A ; b D T AA ;A b C b :...
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