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Unformatted text preview: (c) .a k / l D a kl . (Refer to Appendix C for a discussion of induction and multiple induction.) 2.2.9. Let a be an element of a group. Let n be the least positive integer such that a n D e . Show that e;a;a 2 ;:::;a n ± 1 are all distinct. Conclude that the order of the subgroup generated by a is n . 2.2.10. ˚.14/ is cyclic of order 6. Which elements of ˚.14/ are generators? What is the order of each element of ˚.14/ ? 2.2.11. Show that the order of a cycle in S n is the length of the cycle. For example, the order of .1234/ is 4. What is the order of a product of two disjoint cycles? Begin (of course!) by considering some examples. Note, for instance, that the product of a 2–cycle and a 3–cycle (disjoint) is 6,while the order of the product of two disjoint 2–cycles is 2....
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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, Real Numbers, Multiplication

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