Assignment11Part2

# Assignment11Part2 - Math 302 Modern Algebra Spring 2009...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 302 Modern Algebra Spring 2009 Homework Solutions 7.9 THE SYMMETRIC AND ALTERNATING GROUPS 3. Express as a product of disjoint cycles. (a) µ 1 2 3 4 5 6 7 8 9 2 1 3 5 4 7 9 8 6 ∂ = (1 2)(3)(4 5)(6 7 9)(8) (b) µ 1 2 3 4 5 6 7 8 9 3 5 1 2 4 6 8 9 7 ∂ = (1 3)(2 5 4)(6)(7 8 9) (d) (1 4)(2 7)(5 2 3)(3 4)(1 4 7 2) = (1 5 7 3)(2 4) (e) (7 2 3 6)(8 5)(5 7 1)(1 5 3 7)(4 8 6) = (1 2 3)(4 5 6)(7 8) 4. Write each permutation in Exercise 3 as a product of transpositions. (a) µ 1 2 3 4 5 6 7 8 9 2 1 3 5 4 7 9 8 6 ∂ = (1 2)(4 5)(6 7 9) = (1 2)(4 5)(6 7)(7 9) (b) µ 1 2 3 4 5 6 7 8 9 3 5 1 2 4 6 8 9 7 ∂ = (1 3)(2 5 4)(7 8 9) = (1 3)(2 5)(5 4)(7 8)(8 9) (d) (1 4)(2 7)(5 2 3)(3 4)(1 4 7 2) = (1 5 7 3)(2 4) = (1 5)(5 7)(7 3)(2 4) (e) (7 2 3 6)(8 5)(5 7 1)(1 5 3 7)(4 8 6) = (1 2 3)(4 5 6)(7 8) = (1 2)(2 3)(4 5)(5 6)(7 8) 6. List the elements in each group. (c) A 4 S 4 has 4! = 24 elements, including 2-cycles, 3-cycles, 4-cycles, and products of disjoint 2- cycles. S 4 = { (1) , (1 2) , (1 3) , (1 4) , (2 3) , (2 4) , (3 4) , (1 2 3) , (1 2 4) , (1 3 2) , (1 3 4) , (1 4 2) , (1 4 3) , (2 3 4) , (2 4 3) , (1 2 3 4) , (1 2 4 3) , (1 3 2 4) , (1 3 4 2) , (1 4 2 3) , (1 4 3 2) , (1 2)(3 4) , (1 3)(2 4) , (1 4)(2 3) } . The 3-cycles can be written as products of two transpositions, so are even. The 4-cycles can be written as products of three transpositions, so are odd. Then A 4 contains the three cycles and the products of disjoint 2-cycles: A 4 =...
View Full Document

## This note was uploaded on 10/20/2009 for the course MATH 302 taught by Professor Edwards during the Spring '03 term at CSU Fullerton.

### Page1 / 3

Assignment11Part2 - Math 302 Modern Algebra Spring 2009...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online