221. ALGEBRAIC THEMESContinue in this way untilhas been written as a product of disjoint cy-cles.Let me show you how to express the idea of the algorithm a little moreformally and also more concisely, using mathematical induction.8In thepreceding explanation, the phrase “continue in this way” is a signal that toformalize the argument it is necessary to use induction.Because disjoint cycles1and2commute (12D21, Exercise1.5.13), uniqueness in the following statement means uniqueness up to or-der; the factors are unique, and the order in which the factors are writtenis irrelevant. Also note that.a1; a2; : : : ; ak/is the same cyclic permuta-tion as.a2; : : : ; ak; a1/, and there is no preferred first entry in the cyclenotation. Finally, in order not to have to make an exception for the identityelemente, we regardeas the product of the empty collection of cycles.Theorem 1.5.3.Every permutation of a finite set can be written uniquelyas a product of disjoint cycles.
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