College Algebra Exam Review 13

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online