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Solutions for Modern Algebra: An Introduction | 6th Edition | ISBN: 9780470384435 | Authors: John R. Durbin 9780470384435

Solution for problem 4.5 Chapter 4

(a) Consider Example 4.1, and let a denote clockwise rotation through n /2 radians. Let

Modern Algebra: An Introduction | 6th Edition


Problem 4.5

(a) Consider Example 4.1, and let a denote clockwise rotation through n /2 radians. Let Hdenote the smallest subset of G such that a E H and H is closed with respect to o. DetermineH. (Suggestion: Denote a 0 a by a 2 , denote a 0 a 0 a by a 3 , and so on. H containsfour distinct elements.)(b) Same as (a) with n/2 replaced by n/6.(c) Same as (a) with n/2 replaced by n/k (k EN). How many distinct elements (rotations)does H contain? (Treat rotations as indistinct if they differ by an integral mUltiple of 2n.)

Accepted Solution
Step-by-Step Solution:

Step 1 of 3

(a)

The set  is closed under  since  represents  rotation which is same as initial position. The set  is smallest set since  is generated by .

The clockwise rotation is as follows:

                                                                 

Chapter 4, Problem 4.5 is Solved

Step 2 of 3


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(a) Consider Example 4.1, and let a denote clockwise rotation through n /2 radians. Let