MATH 111 m4

MATH 111 m4 - MODEL ANSWERS TO THE FOURTH HOMEWORK 2...

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Unformatted text preview: MODEL ANSWERS TO THE FOURTH HOMEWORK 2. Chapter 3, Section 1: 1 (a) 1 2 3 4 5 6 4 5 2 1 3 6 . (b) 1 2 3 4 5 3 1 2 4 5 . (c) 1 2 3 4 5 1 4 3 2 5 . 5. It suffices to find the cycle type and take the lowest common multi- ples of the individual lengths of a cycle decomposition. (a) (1 , 4)(2 , 5 , 3) Order 6. (b) (1 , 3 , 2) Order 3. (c) (2 , 4) Order 2. 2. Chapter 3, Section 2: 1 As σ and τ are cycles, we may find integers a 1 , a 2 , . . . , a k and b 1 , b 2 , . . . , b l such that σ = ( a 1 , a 2 , . . . , a k ) and τ = ( b 1 , b 2 , . . . , b l ). To say that σ and τ are disjoint cycles is equivalent to saying that the two sets S = { a 1 , a 2 , . . . , a k } and T = { b 1 , b 2 , . . . , b l } are disjoint. We want to prove that στ = τσ. As both sides of this equation are permutations of the first n natural numbers, it suffices to show that they have the same effect on any integer 1 ≤ j ≤ n ....
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MATH 111 m4 - MODEL ANSWERS TO THE FOURTH HOMEWORK 2...

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