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Math / Modern Algebra: An Introduction 6 / Chapter 4 / Problem 4.5

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

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

Chapter 4, Problem 4.5

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QUESTION:

(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.)

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QUESTION:

(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.)

ANSWER:

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:

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