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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