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Unformatted text preview: 104 2. BASIC THEORY OF GROUPS 2.2.12. Can an abelian group have exactly two elements of order 2? 2.2.13. Suppose an abelian group has an element a of order 4 and an el ement b of order 3. Show that it must also have elements of order 2 and 6. 2.2.14. Suppose that a group G contains two elements a and b such that ab D ba and the orders o.a/ and o.b/ of a and b are relatively prime. Show that the order of ab is o.a/o.b/ . 2.2.15. This exercise generalizes the previous one. Suppose that a group G contains two elements a and b such that ab D ba and h a i\h b i D f e g . (a) Show that if a k b ` D e , then a k D e and b ` D e . (b) Find the order of ab in terms of o.a/ and o.b/ . 2.2.16. Show that the symmetric group S n (for n 2 ) is generated by the 2–cycles .12/;.23/;:::;.n 1 n/ . 2.2.17. Show that the symmetric group S n (for n 2 ) is generated by the 2–cycle .12/ and the n –cycle .12:::n/ ....
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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

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