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College Algebra Exam Review 108

# College Algebra Exam Review 108 - T.µ 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 . .p// D . /.p/ for all and 2 S n 2.4.11. Now fix n 2 N , and define D Y 1 i<j n .x i x j /: For any 2 S n , check that . / D ˙ . Show that the map W 7! . /= 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 is a product of k 2–cycles , then . / D . 1/ k . Now any permutation can be written as a product of 2–cycles (Exercise 1.5.5 ). If a permutation can be written as a product of k 2–cycles and also as a product of l 2–cycles, then . / 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, . / D 1 . 2.4.13. For each permutation 2 S n , define an n -by- n matrix T . / 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 . / D OE O e .1/ ; O e .2/ ; : : : ; O e .n/ Ł I that is, the k th column of T . / is the basis vector O e .k/ . Show that the map 7! T . / is a homomorphism from S n into GL .n;
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Unformatted text preview: T.µ/ is a homomorphism from S n into GL .n; R / . What is the range of T ? Deﬁnition 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 ² 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 µ 2 S n , we have µ.a 1 ;a 2 ;:::;a k /µ ² 1 D .µ.a 1 /;µ.a 2 /;:::;µ.a k //: Hint: As always, ﬁrst look at some examples for small n and k . Both sides are permutations (i.e., bijective maps deﬁned 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 µ 2 S n such that µ.a 1 ;a 2 ;:::;a k /µ ² 1 D .b 1 ;b 2 ;:::;b k /:...
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