hw26 - G c Prove that |h a i| = | a | d Prove that G = h a...

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Intro to Abstract Math Homework 26 Fall 2009 Due November 16 Exercise 75. Let G be a group with identity element e and let a G . Suppose that | a | = n . a. Prove that a k = e if and only if n | k . [ Hint: One implication follows from the laws of exponents. For the other, write k = qn + r , where r Z n is the remainder when k is divided by n . Show that r 6 = 0 contradicts the fact that | a | = n .] b. Prove that e,a,a 2 ,...,a n - 1 are distinct elements of
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Unformatted text preview: G . c. Prove that |h a i| = | a | . d. Prove that G = h a i if and only if | G | = | a | . Exercise 76. Find all the cyclic subgroups of ( Z n , + n ) for n = 4 , 5 , 10 , 12. Identify those a (if there are any) for which Z n = h a i . [ Suggestion: Just compute h a i for every a .] Exercise 77. Find all the cyclic subgroups of S 3 ....
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This note was uploaded on 02/29/2012 for the course MATH 3326 taught by Professor Ryandaileda during the Fall '09 term at Trinity University.

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