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College Algebra Exam Review 12

College Algebra Exam Review 12 - 22 1 ALGEBRAIC THEMES...

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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. Theorem 1.5.3. Every permutation of a finite set can be written uniquely as a product of disjoint cycles.
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