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 conjugate 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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 Fall '08
 EVERAGE
 Algebra

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