College Algebra Exam Review 14

College Algebra Exam Review 14 - 24 1. ALGEBRAIC THEMES...

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24 1. ALGEBRAIC THEMES 1.5.8. Show that the multiplication in S n is noncommutative for all n ± 3 . Hint: Find a pair of 2–cycles that do not commute. 1.5.9. Let ± n denote the perfect shuffle of a deck of 2n cards. Regard ± n as a bijective function of the set f 1;2;:::;2n g . Find a formula for ± n .j/ , when 1 ² j ² n , and another formula for ± n .j/ , when n C 1 ² j ² 2n . 1.5.10. Explain why a cycle of length k has order k . Explain why the order of a product of disjoint cycles is the least common multiple of the lengths of the cycles. Use examples to clarify the phenomena for yourself and to illustrate your explanation. 1.5.11. Find the cycle decomposition for the perfect shuffle for decks of size 2, 4, 6, 12, 14, 16, 52. What is the order of each of these shuffles? 1.5.12. Find the inverse, in two–line notation, for the perfect shuffle for decks of size 2, 4, 6, 8, 10, 12, 14, 16. Can you find a rule describing the inverse of the perfect shuffle in general? The following two exercises supply important details for the proof of
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