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By Frederick M. Goodman

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Additional resources for Algebra: Abstract and Concrete (Stressing Symmetry) (2.5 Edition)

Example text

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. 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? 12. Find the inverse, in two–line notation, for the perfect shuffle for decks of size 2, 4, 6, 8, 10, 12, 14, 16.

4. Compute the congruence class modulo 12 of 4237 . 10 constitute an experimental investigation of zero divisors and invertible elements in Zn . 5. Can an element of Zn be both invertible and a zero divisor? 6. If an element of Zn is invertible, is its multiplicative inverse unique? That is, if Œa is invertible, can there be two distinct elements Œb and Œc such that ŒaŒb D Œ1 and ŒaŒc D Œ1? 7. If a nonzero element Œa of Zn is a zero divisor, can there be two distinct nonzero elements Œb and Œc such that ŒaŒb D Œ0 and ŒaŒc D Œ0?

K 1/ 2-cycles. Conclude that any permutation can be written as a product of some number of 2-cycles. 3 to discover the right pattern. Then do a proper proof by induction. 6. Explain how to compute the inverse of a permutation that is given in two-line notation. 7. Explain how to compute the inverse of a permutation that is given as a product of cycles (disjoint or not). One trick of problem solving is to simplify the problem by considering special cases. First you should consider the case of a single cycle, and it will probably be helpful to begin with a short cycle.

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