College Algebra Exam Review 13

# College Algebra Exam Review 13 - of some number of 2-cycles...

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1.5. PERMUTATIONS 23 Exercises 1.5 1.5.1. Work out the full multiplication table for the set of permutations of three objects. 1.5.2. Compare the multiplication table of S 3 with that for the set of sym- metries of an equilateral triangular card. (See Figure 1.3.6 on page 10 and compare Exercise 1.3.1 .) Find an appropriate matching identiﬁcation or matching of elements of S 3 with symmetries of the triangle that makes the two multiplication tables agree. 1.5.3. Work out the decomposition in disjoint cycles for the following: (a) ± 1 2 3 4 5 6 7 2 5 6 3 7 4 1 ² (b) .12/.12345/ (c) .14/.12345/ (d) .12/.2345/ (e) .13/.2345/ (f) .12/.23/.34/ (g) .12/.13/.14/ (h) .13/.1234/.13/ 1.5.4. On the basis of your computations in Exercise 1.5.3 , make some conjectures about patterns for certain products of 2-cycles, and for certain products of two-cycles and other cycles. 1.5.5. Show that any k -cycle .a 1 ;:::;a k / can be written as a product of .k ± 1/ 2-cycles. Conclude that any permutation can be written as a product
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Unformatted text preview: of some number of 2-cycles. Hint: For the ﬁrst part, look at your compu-tations in Exercise 1.5.3 to discover the right pattern. Then do a proper proof by induction. 1.5.6. Explain how to compute the inverse of a permutation that is given in two-line notation. Compute the inverse of ± 1 2 3 4 5 6 7 2 5 6 3 7 4 1 ² : 1.5.7. 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. A 2-cycle is its own inverse, so the ﬁrst interesting case is that of a 3-cycle. Once you have ﬁgured out the inverse for a 3-cycle and a 4-cycle, you will probably be able to guess the general pattern. Now you can begin work on a product of several cycles....
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