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Unformatted text preview: 22 1. ALGEBRAIC THEMES Continue in this way until has been written as a product of disjoint cy cles. Let me show you how to express the idea of the algorithm a little more formally and also more concisely, using mathematical induction. 8 In the preceding explanation, the phrase continue in this way is a signal that to formalize the argument it is necessary to use induction. Because disjoint cycles 1 and 2 commute ( 1 2 D 2 1 , Exercise 1.5.13 ), uniqueness in the following statement means uniqueness up to or der; the factors are unique, and the order in which the factors are written is irrelevant. Also note that .a 1 ;a 2 ;:::;a k / is the same cyclic permuta tion as .a 2 ;:::;a k ;a 1 / , and there is no preferred first entry in the cycle notation. Finally, in order not to have to make an exception for the identity element e , we regard e as the product of the empty collection of cycles....
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra, Mathematical Induction

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