By F. Goodman

ISBN-10: 0130673420

ISBN-13: 9780130673428

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**Sample text**

Bookmt” — 2006/8/8 — 12:58 — page 24 — #36 ✐ 24 ✐ 1. 8. Show that the multiplication in Sn is noncommutative for all n Hint: Find a pair of 2–cycles that do not commute. 3. 9. Let n denote the perfect shuffle of a deck of 2n cards. Regard n as a bijective function of the set f1; 2; : : : ; 2ng. j /, when n C 1 Ä j Ä 2n. 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.

Thus the product is 13 4765 1423 56 D 17642 : Notice that the permutation D 1 4 2 3 5 6 is the product of the cycles 1 4 2 3 and 5 6 . Disjoint cycles commute; their product is independent of the order in which they are multiplied. For example, Â Ã 1 2 3 4 5 6 7 Œ 1 4 2 3 Œ 5 6 D Œ 5 6 Œ 1 4 2 3 D : 4 3 1 2 6 5 7 A permutation is said to have order k if the k t h power of is the identity and no lower power of is the identity. A k-cycle (that is, a cycle of length k) has order k. For example, 2 4 3 5 has order 4.

Now you can begin work on a product of several cycles. ✐ ✐ ✐ ✐ ✐ ✐ “bookmt” — 2006/8/8 — 12:58 — page 24 — #36 ✐ 24 ✐ 1. 8. Show that the multiplication in Sn is noncommutative for all n Hint: Find a pair of 2–cycles that do not commute. 3. 9. Let n denote the perfect shuffle of a deck of 2n cards. Regard n as a bijective function of the set f1; 2; : : : ; 2ng. j /, when n C 1 Ä j Ä 2n. 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.

### Algebra. Abstract and Concrete by F. Goodman

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