College Algebra Exam Review 108

College Algebra Exam Review 108 - T./ is a homomorphism...

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118 2. BASIC THEORY OF GROUPS 2.4.10. Let f x 1 ;x 2 ;:::;x n g be variables. For any polynomial p in n vari- ables and for ± 2 S n , define ±.p/.x 1 ;:::;x n / D p.x ±.1/ ;:::;x ±.n/ /: Check that ±.².p// D .±²/.p/ for all ± and ² 2 S n 2.4.11. Now fix n 2 N , and define ³ D Y 1 ± i<j ± n .x i ± x j /: For any ± 2 S n , check that ±.³/ D ˙ ³ . Show that the map ´ W ± 7! ±.³/=³ is a homomorphism from S n to f 1; ± 1 g . 2.4.12. Show that for any 2–cycle .a;b/ , ´..a;b// D ± 1 ; hence if a per- mutation µ is a product of k 2–cycles , then ´.µ/ D . ± 1/ k . Now any permutation can be written as a product of 2–cycles (Exercise 1.5.5 ). If a permutation µ can be written as a product of k 2–cycles and also as a product of l 2–cycles, then ´.µ/ D . ± 1/ k D . ± 1/ l , so the parity of k and of l is the same. The parity is even if, and only if, ´.µ/ D 1 . 2.4.13. For each permutation µ 2 S n , define an n -by- n matrix T.µ/ as follows. Let O e 1 ; O e 2 ;:::; O e n be the standard basis of R n ; O e k has a 1 in the k th coordinate and zeros elsewhere. Define T.µ/ D Œ O e ².1/ ; O e ².2/ ;:::; O e ².n/ Ł I that is, the k th column of T.µ/ is the basis vector O e ².k/ . Show that the map µ 7!
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Unformatted text preview: T./ is a homomorphism from S n into GL .n; R / . What is the range of T ? Denition 2.4.25. Two elements a and b in a group G are said to be con-jugate if there is an element g 2 G such that a D gbg 1 . 2.4.14. This exercise determines when two elements of S n are conjugate. (a) Show that for any k cycle .a 1 ;a 2 ;:::;a k / 2 S n , and for any permutation 2 S n , we have .a 1 ;a 2 ;:::;a k / 1 D ..a 1 /;.a 2 /;:::;.a k //: Hint: As always, rst look at some examples for small n and k . Both sides are permutations (i.e., bijective maps dened on f 1;2;:::;n g ). Show that they are the same maps by showing that they do the same thing. (b) Show that for any two k cycles, .a 1 ;a 2 ;:::;a k / and .b 1 ;b 2 ;:::;b k / in S n there is a permutation 2 S n such that .a 1 ;a 2 ;:::;a k / 1 D .b 1 ;b 2 ;:::;b k /:...
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