4023f11ex3

# 4023f11ex3 - (a) Determine the orbit, Orb( x ) for x = 1...

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

Name: Exam 3 Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Put your name on each page of your paper. 1. [15 Points] Let = ± 1 2 3 4 5 6 2 1 3 5 4 6 and = ± 1 2 3 4 5 6 6 1 2 4 3 5 . Compute the following permutations: (a) ﬁﬂ (b) ﬂﬁ (c) ¡ 1 2. [12 Points] Let ± 2 S 10 be the permutation ± 1 2 3 4 5 6 7 8 9 10 9 7 8 4 3 2 10 5 1 6 : (a) Find the cycle decomposition of ± . (b) Determine if ± is even or odd. 3. [12 Points] How many permutations in S 7 have cycle structure (1 ; 2 ; 2 ; 2)? 4. [12 Points] What is the number of elements in each of the following groups? (a) S n (c) D n (c) C n 5. [15 Points] Let G be the following group of permutations of the set X = f 1 ; 2 ; 3 ; 4 ; 5 g : G = ' (1) ; ¡ 1 4 ¢¡ 2 5 3 ¢ ; ¡ 2 3 5 ¢ ; ¡ 1 4 ¢ ; ¡ 2 5 3 ¢ ; ¡ 2 3 5 ¢¡ 1 4 ¢“ = ' e; ±; ± 2 ; ± 3 ; ± 4 ; ± 5 : Thus G acts on X via these permutations.
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: (a) Determine the orbit, Orb( x ) for x = 1 and x = 2. (b) Determine the stabilizer subgroup Stab( x ) for x = 1 and x = 2. (c) Use your results to verify that j Orb( x ) j = j G j = j Stab( x ) j for x = 1 and x = 2. 6. [9 Points] Fill in the missing parts to complete the statement of Burnside’s Theorem: If G is a ﬂnite group that acts on a ﬂnite set X , then the number k of distinct orbits is: k = 1 X : 7. [25 Points] Find the number of patterns obtained by coloring the vertices of a regular pentagon with 4 colors. Be sure to explain your calculations, including identiﬂcation of the symmetry group and all of the ﬂxed point calculations needed. Math 4023 November 21, 2011 1...
View Full Document

## This document was uploaded on 12/28/2011.

Ask a homework question - tutors are online