4023f11ps8

# 4023f11ps8 - 4 Deﬂne ² Z n Z n by ² x = 3 x(a If gcd(3...

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Homework #8 Due: November 9, 2011 1. Factor each of the following permutations into disjoint cycles. Using your answer express the inverse as a product of disjoint cycles and determine if each permutation is even or odd. (a) ± 1 2 3 4 5 6 7 8 9 4 7 9 8 2 1 6 3 5 (b) ± 1 2 3 4 5 6 7 8 9 6 4 8 9 3 1 7 5 2 (c) ¡ 1 3 ¢¡ 2 5 7 ¢¡ 3 8 5 ¢ (d) ¡ 3 5 7 ¢¡ 2 3 4 7 6 ¢¡ 3 5 7 ¢ ¡ 1 2. (a) How many permutations in S 5 ﬂx 1? (b) How many permutations in S 5 ﬂx both 1 and 3? 3. Give an example of , , ± 2 S 5 , none of which is the identity, with ﬁﬂ = ﬂﬁ and ﬁ± = ±ﬁ , but with ﬂ± 6 = ±ﬂ
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Unformatted text preview: . 4. Deﬂne ² : Z n ! Z n by ² ( x ) = 3 x . (a) If gcd(3 ; n ) = 1 show that ² is a permutation of Z n . (b) If n = 7 verify that the permutation ² is a cycle of length 6. (c) If n = 10, ﬂnd the disjoint cycle factorization of ² . 5. Compute ﬁ ¡ 1 , ﬁ 2 , ﬁ 3 , ﬁ 4 , ﬁ 5 where (a) ﬁ = ¡ 1 2 3 ¢ (b) ﬁ = ¡ 1 2 3 4 ¢ (c) ﬁ = ¡ 1 2 3 4 5 6 ¢ : 6. How many elements of S 10 have the cycle structure (1 ; 4 ; 5)? How many have the cycle structure (2 ; 2 ; 2 ; 4)? Math 4023 1...
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## This document was uploaded on 12/28/2011.

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